Say a mad scientist detonated a nuclear device that killed everyone in Washington including everyone in the president's line of succession and say, he somehow figured out where the designated survivor was and killed him also.
What would happen then? How would the government decide who is president or who would be making the decisions?
The states would hold special elections to replace their members of Congress who got killed in the blast. The House of Representatives would then elect a new Speaker, who would assume the presidency.
If most of Congress remains intact, then Dan's answer is correct, as the first person to get appointed Speaker of the House or President pro tempore of the senate will assume the role of President. And as long as the House has enough members to be (constitutionally) functional, I expect the Senate to let the House do their appointment first (as the House's composition more closely reflects the overall population). However, as the OP supposes that essentially the entire capital is wiped out, it seems better to suppose that all of the members of both chambers are dead. In which case I rather disagree with Dan's answer, as it takes entirely too long to constitutionally resolve the crisis of an empty Presidency and empty everything else if you go through the House.
The problem is that the constitution requires that members of the House of Representatives be voted in by popular elections (Article 1, Section 2, Clause 1).
The House of Representatives shall be composed of Members chosen every second Year by the People of the several States...
This applies even to vacancies, which you'd have in this situation (Clause 4):
When vacancies happen in the Representation from any State, the Executive Authority thereof shall issue Writs of Election to fill such Vacancies.
So even to fill vacancies in the House, full and proper elections must be held. There's no way of getting around this if you want it to be done constitutionally.
However, Senators can be appointed when there are vacancies, depending on State laws (the 17th Amendment):
When vacancies happen in the representation of any State in the Senate, the executive authority of such State shall issue writs of election to fill such vacancies: Provided, That the legislature of any State may empower the executive thereof to make temporary appointments until the people fill the vacancies by election as the legislature may direct.
Now elections are not simple things to put together. The UK currently expects that it would take a year or more to put together a second Brexit referendum for example. Even trying to expedite the process you're probably still looking at weeks or more to start filling up the House.
However, Senate positions could be filled within hours or less. Any states which do not currently empower their Governors (aka 'executive authority') to make appointments could call an emergency session to grant them this power (even temporarily), and any Governor so empowered would then quickly make appointments that are then likely to be quickly approved (possibly before that first emergency session has even ended; the legislatures could just twiddle their thumbs until the Governor gives them the appointments).
In any case, all you need is for enough Senators to show up and hold a session to elect a new President Pro Tempore, who would then immediately become President of the United States according to the line of succession (as the House still needs time to cobble together enough people to elect a Speaker). All we need is to satisfy that "enough senators show up" part.
Current Supreme Court precedent says that the Senate has the necessary quorum as long as its rules say it does, and that the court does not question if the Senate has a quorum or not. The Senate rules presume a quorum exists by default, and it's only if a member explicitly invokes a quorum check that an actual count is made. What this means here is that as long as nobody in this session questions if the body has a quorum, then it is presumed to have one and it can carry out any business it is capable of doing with the people there. So you don't even need a lot of new Senators to have been appointed in. In principle a handful or two is all you need, though one might imagine that at least dozens would be allowed to assemble first in order to increase the perception of legitimacy without unduly delaying filling the office of the President.
However, it may not be immediately apparent that existing SCOTUS precedents cover such an extreme case, so one may still desire that the quorum be strictly met to ensure legitimacy. But this should be only a minor obstacle, presenting mostly increased logistical challenges in safely transporting new Senators to the (interim) Senate chambers. A true quorum requires 51 of the 100 possible Senators to be present. That requires a full set of appointees from at most 26 states to be satisfied. Currently, 36 states already allow gubernatorial appointments to vacant Senate seats, and 9 more allow interim appointments under certain situations.* That gives a lot of wiggle room, so if there would still be political fights in some states over the appointments (such a catastrophe would tend to be unifying, though) we should still be able to achieve 51 or more new Senators in relatively short order. And if things go smoothly we'd have 70-90 newly appointed Senators without any legislatures having to change their laws.
And that's that. Once the President Pro Tempore of the Senate exists, he becomes President of the United States, and there is no longer any need to invoke the line of succession. By 3 U.S.C. § 19(c) the acting President will remain in that role until the next election. And it is easily the fastest way of filling a position in the line of succession in such an extreme situation. It could conceivably be done in a timescale of hours, with quite a lot of that being waiting for new appointees to be flown in to what will serve as the Senate chambers, especially given that the Military will probably be heavily involved in an attempt to ensure their safety.
(See also Ohwilleke's answer for a more thorough discussion of the legal basis)
*The link also provides details on how states that require an election (eventually) structure that requirement. Most of them allow for several months to pass. And in some situations and states this is apparently required. Though some require elections within 30 days.
If you kill enough leaders to destablise a country the military will probably take power and try to defend the country against whoever is attacking it. They might also try to keep public order.
Afterwards the military would hopefully try to oversee a return to the status quo where proper elected officials hold power.
The next President would be either the Speaker of the U.S. House, following a re-election of enough seats in Congress to have a quorum to fill that post, or the Speaker Pro Temp of the U.S. Senate, following appointment of enough people to the U.S. Senate to have a quorum of enough Senators to elect someone to that post.
Realistically, given that Governors can appoint replacement U.S. Senators more quickly than representatives to the U.S. House can be elected in special elections, the position of Speaker Pro Temp of the U.S. Senate would be filled first, and that person would become the Acting President.
The answer from @zibadawatimmy is close to correct, but doesn't fully spell out the legal authorities that support that answer, so I do so more fully in this answer.
This follows from U.S. Constitution, Article II, Section 1, Clause 6 (as amended by U.S. Constitutional Amendment XXV in respects irrelevant to this question) which authorized Congress to enact a Presidential succession law. It states:
In the Case of the Removal of the President from Office, or of his Death, Resignation, or Inability to discharge the Powers and Duties of the said Office, the Same shall devolve on the Vice President, and Congress may be Law provide for the Case of Removal, Death, Resignation or Inability, both of the President and Vice President, declaring what Officer shall then act as President, and such Officer shall act accordingly, until the Disability be remove, or a President shall be elected.
The Presidential Succession Act of 1947, as amended is the version currently in force at Title 3 of the United States Code, Section 19 (see also this Wikipedia article on the topic which also summarizes the history of all actual permanent and temporary Presidential successions in the U.S.). The Presidential Succession Act states:
(a)(1) If, by reason of death, resignation, removal from office, inability, or failure to qualify, there is neither a President nor Vice President to discharge the powers and duties of the office of President, then the Speaker of the House of Representatives shall, upon his resignation as Speaker and as Representative in Congress, act as President.
(2) The same rule shall apply in the case of the death, resignation, removal from office, or inability of an individual acting as President under this subsection. (b) If, at the time when under subsection (a) of this section a Speaker is to begin the discharge of the powers and duties of the office of President, there is no Speaker, or the Speaker fails to qualify as Acting President, then the President pro tempore of the Senate shall, upon his resignation as President pro tempore and as Senator, act as President.
(c) An individual acting as President under subsection (a) or subsection (b) of this section shall continue to act until the expiration of the then current Presidential term, except that—
(1) if his discharge of the powers and duties of the office is founded in whole or in part on the failure of both the President-elect and the Vice-President-elect to qualify, then he shall act only until a President or Vice President qualifies; and
(2) if his discharge of the powers and duties of the office is founded in whole or in part on the inability of the President or Vice President, then he shall act only until the removal of the disability of one of such individuals.
(d)(1) If, by reason of death, resignation, removal from office, inability, or failure to qualify, there is no President pro tempore to act as President under subsection (b) of this section, then the officer of the United States who is highest on the following list, and who is not under disability to discharge the powers and duties of the office of President shall act as President: Secretary of State, Secretary of the Treasury, Secretary of Defense, Attorney General, Secretary of the Interior, Secretary of Agriculture, Secretary of Commerce, Secretary of Labor, Secretary of Health and Human Services, Secretary of Housing and Urban Development, Secretary of Transportation, Secretary of Energy, Secretary of Education, Secretary of Veterans Affairs, Secretary of Homeland Security.
(2) An individual acting as President under this subsection shall continue so to do until the expiration of the then current Presidential term, but not after a qualified and prior-entitled individual is able to act, except that the removal of the disability of an individual higher on the list contained in paragraph (1) of this subsection or the ability to qualify on the part of an individual higher on such list shall not terminate his service.
(3) The taking of the oath of office by an individual specified in the list in paragraph (1) of this subsection shall be held to constitute his resignation from the office by virtue of the holding of which he qualifies to act as President. (e) Subsections (a), (b), and (d) of this section shall apply only to such officers as are eligible to the office of President under the Constitution. Subsection (d) of this section shall apply only to officers appointed, by and with the advice and consent of the Senate, prior to the time of the death, resignation, removal from office, inability, or failure to qualify, of the President pro tempore, and only to officers not under impeachment by the House of Representatives at the time the powers and duties of the office of President devolve upon them.
(f) During the period that any individual acts as President under this section, his compensation shall be at the rate then provided by law in the case of the President.
The fair, and I would say, widely accepted reading of the statute, is that if, everyone in 3 U.S.C. § 19(a) and 3 U.S.C. § 19(d)(1) to fill the position, that the first person coming into being in a position designed by 3 U.S.C. § 19(a) and 3 U.S.C. § 19(d)(1) would become the Acting President.
Since no one in 3 U.S.C. § 19(d)(1) can be appointed without a Presidential nomination, the next person to come into being will always be one of the two people designated in 3 U.S.C. § 19(a), i.e. the Speaker of the U.S. House or the President Pro Temp of the U.S. Senate. These posts would be filled as soon as a quorum was present in the U.S. House or the U.S. Senate to elect such a person from their ranks as a first order of business.
Vacancies in the U.S. House are filled by Special Election. U.S. Constitution, Article I, Section 2, Clause 4 ("When vacancies happen in the Representation from any State, the Executive Authority thereof shall issue Writs of Election to fill such Vacancies.").
Vacancies in the U.S. Senate are filled by the Governor of the state with the vacancy. U.S. Constitution, Article I, Section 3, Clause 3, as amended by U.S. Constitutional Amendment XVII. The 17th Amendment states in Clause 2:
When vacancies happen in the representation of any State in the Senate, the executive authority of such State shall issue writs of election to fill such vacancies: Provided, That the legislature of any State may empower the executive thereof to make temporary appointments until the people fill the vacancies by election as the legislature may direct.
Most U.S. States (if not all) have, in fact, empowered their Governor to make temporary appointments pending special elections to the U.S. Senate pursuant to the 17th Amendment (at the time of this Answer 45 state governors have this power, which is sufficient to appoint a quorum; Governors cannot fill U.S. Senate vacancies in ND, OK, OR, RI and WI).
The quorum in both the U.S. House and the U.S. Senate is a majority (i.e. 51 in the U.S. Senate and 218 in the U.S. House).
Each House shall be the Judge of the Elections, Returns and Qualifications of its own Members, and a Majority of each shall constitute a Quorum to do Business; but a smaller Number may adjourn from day to day, and may be authorized to compel the Attendance of absent Members[.]
U.S. Constitution, Article I, Section 5, Clause 1.
The process of Governors appointing U.S. Senators is faster than the process of conducting a special election, so a quorum of U.S. Senators would exist and appoint a President Pro Temp more quickly than the U.S. House would attain a quorum and appoint a Speaker of the House. Once that President Pro Temp was elected, that person would become acting President pursuant to 3 U.S.C. § 19(a)(2), that Senator's seat would become vacant and have to be filled again, and the U.S. Senate would elect a new President Pro Temp.
The since the Acting President was either a Speaker of the House of President Pro Temp of the Senate appointed pursuant to 3 U.S.C. § 19(a), this person would serve of the remainder of the currently Presidential term pursuant to 3 U.S.C. § 19(c).
So everybody in Washington is dead. It would be an unprecedented crisis to be sure.
The first thing that would happen would vary from state to state. Most states have a provision to allow the governor to appoint people to Congressional positions until a special election is held. For those states without such a provision, my bet is those state legislatures would quickly enact an exception to allow rapid appointment of replacements, lest those states find themselves without representation. If they didn't, Congress would convene once they had a quorum (51 Senators and 218 House Representatives), and there's enough states with executive appointment to meet that threshold.
While it's not clear on the Presidency (again, everyone being dead is unprecedented), the safest bet here is that the House would invoke Article II of the Constitution and select a new President
and if there be more than one who have such Majority, and have an equal Number of Votes, then the House of Representatives shall immediately choose by Ballot one of them for President
No vote would have taken place, of course, but there's no legitimate successor. Having the House use its defined role to do this makes the most sense, since it is the most expedient way to fill the role (which is priority #1). This could be the Speaker, but my bet is the majority party will select someone from their own party (probably a former candidate or cabinet member, since most do not live in Washington). At this point, the Senate would likely move to quickly confirm the new VP and cabinet positions in the White House. Congress would probably pass a few other necessary bills to keep the government running and then adjourn, since every member of Congress now has a special election to prepare for.
The newly appointed President would then serve the remainder of the term of their predecessor.
