They should have been forced to roll the initial save regardless
The wizard cast Eyebite, which says (emphasis mine):
One creature of your choice within 60 feet of you that you can see must succeed on a Wisdom saving throw or be affected by one of the following effects of your choice for the duration.
The wizard selected Asleep. At this point the assassin rolls a Wisdom save, as directed. If the assassin makes the initial save, they are not affected the current round, and cannot be a future target because the wizard:
can't target a creature again if it has succeeded on a saving throw against this casting of eyebite.
However, if the assassin failed the initial save, then the Asleep effect is applied to them. At that point it has no result because of their fey ancestry.
See also: Does a creature that is immune to all outcomes (conditions, damage, etc.) of an effect still make a saving throw? and Does Calm Emotions still allow a saving throw?
Now the assassin is under the effects of the Eyebite spell, but the specific asleep effect is 'suppressed'. If something removed their fey ancestry (for example, polymorph) then the asleep would take effect. The spell has not ended, since its duration is Concentration / 1 minute, and the suppressed effect will persist until the end of the spell.
If the wizard then wishes to try Sickened on the next round, they may.
On each of your turns until the spell ends, you can use your action to target another creature but can't target a creature again if it has succeeded on a saving throw against this casting of eyebite.
The 'on each of your turns' says you can target 'another' creature, which by itself might imply that you can't target the same creature again. However, by then saying you can't target the same creature again if it succeeded on its save, the spell clarifies that you can re-target the same creature if it failed.
When the wizard again targets the assassin:
One creature of your choice within 60 feet of you that you can see must succeed on a Wisdom saving throw or be affected by one of the following effects of your choice for the duration.
So the assassin now gets a new, second save. Success on this will mean that they are not sickened and cannot be targeted again. Failure will mean that they are sickened. But in either case, success or failure, they will still be carrying the suppressed asleep effect until the end of the spell's duration.