I think "good thing we came to the hospital" is a valod translation, even if it is a bit colloquial.
The ending of "geldik" makes it clear that the subject is "biz". For the subjects ben/sen/o/biz/siz/onlar it would be geldim/geldin/geldi/geldik/geldiniz/geldiler.
It rejected 'luckily we came to the hospital' and 'fortunately we came to hospital' both of which should be accepted. Luckily is obviously a synonym of fortunately, and in British English we leave out the article with places like hospital, church, school. To be in hospital, to go to church, to come to school, etc. Go to hospital and come to hospital are completely normal and acceptable in BE.
It's in the "-k" at the end of the verb.
Look at the conjugation of gelmek in the past tense:
- geldim (I came)
- geldin (You came)
- geldi (he/she came)
- geldik (we came)
- geldiniz (you came)
- geldiler (they came)