There are two places that can make the doctor fly. Maybe more- an hour ago I was only aware of one :)
Since the game was rushed a bit, there's not really much subtlety to the tactics here. You need to be fast with the minigames, try to remember which of the rooms are busy, so you don't deliver a patient there, and make good decisions in patient vs skeleton situation, basically taking into account the time patient has left.
Thanks for playing!