I suppose that argument could work, if the nature of the child's disease was explained.
You wanted to know which disease she had specifically ? What difference would that make ?
None, as Kirk was brought BACK FROM THE DEAD, rendering the poorly kid plot point entirely redundant. You could lose that aspect and lose nothing as McCoy injects the tribble in an entirely unrelated
If it had been brought up as a reason as to why
McCoy decided to inject the tribble with Khan's blood it would have been pretty clever but as it stands you have one scene which sets up nothing and another which comes out of nowhere.
It does not logically follow that if Khan can save a poorly child he can resurrect the dead.