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.
I think the closest you can come to making that distinction in Turkish is by saying "İyi olan hastaneye geldik." That suggests that among several choices, we came to the hospital that is the good one. In English I consider the statements "we came to the good hospital" and "we came to the hospital that is good" to be fairly synonymous, so you may want to clarify the distinction.