In English, there is certainly a difference between 'what disease' and 'which disease'. The latter implying that there is a limited number of diseases to choose from. Is that the case in Dutch too? Or do you never uses 'wat ziekte'?
That difference doesn't exist in Dutch, wat ziekte is never used. Both what disease and which disease translate to welke ziekte.
Although you could say "wat voor soort ziekte" ("what kind/sort of disease") which you can shorten to "wat voor ziekte" which to me is pretty much the same as "what disease".
you are correct, there no such a thing that you can only use WELKE.