The prince was already suspicious of her. I'm not sure if he had noticed her eyes or not in the battle or her perhaps her frame but with her hand being injured it should be easy for him to put it together now.
My first thought when she slipped away without him noticing was that he couldn’t hear her footsteps and that tipped him off. I find this to be a hilarious detail for some reason. Lol
Well unexpected always happen on the job, what else do you expect her to do? This isn't like today where you can conceal a cut with cosmetic. Also, I think she wasn't outed by the physician, since she covered her bandaged hand in the blanket during the checkup.
pretty sure the hand is gonna give her away. Not that I mind lol. She should just ditch her own king and work for this one instead, even if undercover.