Translation:The children who are in the hospital are not happy.
That would be the translation of: Çocuklar hastanede mutlu değiller. And that doesn't exactly mean the same. It sounds like children aren't happy to be in the hospital. If they were elsewhere, they'd be happier.
The original sentence has no such hint. It just says that thr children are unhappy, and they happen to be in the hospital. We don't know if they're unhappy because of that.
As Alex so nicely laid out to you, there is a not so unimportant difference between the two. I personally am happy that we are only getting points for good translations so we can learn to express ourselves correctly, and not just somehow understandably. Happy learning or not unhappy learning, that is the question. ;-)
I propose "The hospitalised children are not happy" as an alternative to the current translation. The introduction to this unit presented "-ki" to turn a noun into an adjective. Here, the English version is using a subordinate clause instead of an adjective (or may be not - my English is not native).
The children who are in the hospital -- patients The children who are at the hospital -- visiting I don't speak American English, so I don't know if these prepositions have the same functions for Americans.
Is there any difference when writing Turkish? Both in and at are accepted as correct answers.