IMO, using water from a cycled tank will do little to prevent a cycle in the new tank. The bacteria colonize surfaces - live rock, live sand, bioballs, biowheels...little, IMO, is in the water itself. Not enough, IMO, to prevent an ammonia issue in such a small tank with a sudden huge increase in bioload. I would watch the water quality in the hospital very quickly.
If there are a lot of fish, it is better, IMO, to set up a large hospital tank or multiple small one's. A hospital system may already seem stressful and to take a lot of stressed sick fish (which is why they have ick) and put them into even smaller quarters is a potential problem.
How old was your original tank? When did you get these fish? What specifically are they?