If you have doctor level X, we assign an injury in a non-linear way weighted strongly towards shorter injuries between 0 and (10+X)/2 weeks, and then we subtract X/2 weeks from the injury length. So the better your doctor, the higher the odds that when those weeks are subtracted, you'll get an immediately-healed player.
First thing that comes to mind: implement a structure, where a player can play hurt.
Very rough example: put in additional variable, say "health". Injuries affect health, which then recovers towards its maximum value. At your own risk, you can field a player at less than 100%, but he will perform worse and this will slow down his recovery and the benefit he gets from training.
This opens the door for new types of staff like rehab specialists, and whatnot. Thoughts, everyone?
"I don't know half of you half as well as I should like; and I like less than half of you half as well as you deserve."