"Where are you going from the hospital?"
Translation:Hastaneden nereye gidiyorsun?
Because the word is spelt "hastane" according to TDK. You use "hane" when the previous word ends in a consonant I guess, like: dershane, tophane, darphane and so on. Otherwise just add "-ne": Postane, pastane.
I don't agree with your logic. deorme90 is more correct. The original word is hastahane - hasta - hane, not hast hane. Hastahane may not be "modern" Turkish, but that doesn't make it wrong.
It is not MY logic. TDK spells it that way, which makes it correct and yours incorrect.
And by the way, with your logic, we should also write: Kahve altı instead of "kahvaltı", because obviously it comes from kahve and altı.
Same thing with "Pazar ertesi" → "Pazartesi".
Spelling doesn't have to reflect the origins of a word.
hastahane is never used in modern (at least written) turkish although it might be the 'original' version.
NereDe - where (at).
NereYe - where (to).
for example, "Sen nereDesin?" is "(at) where are you," whereas "NereYe gidiyorsun?" is "(to) where are you going?"