The best thing to do is to normalise it. Have some doctor friends come and visit and maybe talk about a patient - completely fictitious! - that has a similar problem. Play Doctors and Nurses and let your daughter be the doctor.
DS for some reason loves going to A&E. He has been really ill and still loves it. I don't know why. He also loves our GP and the local nurses (who change dressings, clean wounds and jab needles in). I think because they smile and talk to him - and he's seen them at my appointments too. I'm very relaxed about it: even with a scald (freshly brewed black coffee here too: the best thing to scald yourself with btw as it's cleanest and cools down fastest), I was calm. That helps a lot. I know it's hard to relax when your child is injured, but getting scared yourself doesn't help.
The local A&E department also hand out teddy bears to all children who visit. From the Freemasons apparently.
DS likes that too.