Firstly sorry to hear of your accident. I can't imagine much worse then being injured due to not your own fault.
However regarding compensation specifically I think you're getting into complex legal augments about liability there. That's what the previous posters are reacting to.
Basically should the employer reasonably have known the door was faulty. Did they provide you the right training and supervision for your checks.
Presumably your checks are the process by which they find out about problems with the fire door, so was the damage by wear and tear, or did someone do some repairs recently and not fit the handle according to a reasonable standard. Or were the checks not happening previously? But even if they weren't, would it have been discovered without injury?
As you can see, it's extremely hard to work out if anyone is at fault, but you needn't worry about that too much now.
However, having said all that, since it happened at work, I'm pretty sure your company have to give you some benefits still regardless of liability. Check their hr policies if they have them, if not get advice. There's no time limit that's super urgent, but all your physiotherapy and whatnot are a business expense in effect. You shouldn't be left out of pocket.
Good luck and get well soon!