What was it she died of, the clap?
The example sentence is fine. The question portion (in italics) of it is in the form of an interrogative it-cleft.
Consider a simple declarative:
- She died of [the clap].
and then, a possible it-cleft:
- It was [the clap](i) [(that) she died of __(i)].
and now, convert the main clause into an in situ interrogative clause:
- It was what(i) [(that) she died of __(i)]?
and now front the interrogative word:
- What(i) was it [(that) she died of __(i)]?
You will see that #4 is the same as the question portion of your original example:
What was it she died of, the clap?
ALTERNATIVELY: It could also be looked at in a way similar to the way you were doing it.
Consider an in situ interrogative clause:
B1. She died of what?
and then, into an it-cleft:
B2. It was what(i) [(that) she died of __(i)]?
and now front the interrogative word:
B3. What(i) was it [(that) she died of __(i)]?
which again is the same as the question portion of your original example:
What was it she died of, the clap?
So, you can do it either way. :)