When I checked on her this morning she seemed to be trying on her own and was kinda flopping about. I let her keep trying and this afternoon she is walking a bit but favoring one leg. She seems to be regaining the ability to move her claws. One leg is almost normal and the other is nowhere near as bad as yesterday. I think I'm going to let her continue mending on her own. She doesn't seem to be in pain and she doesn't mind me moving her legs or claws. If she's not walking correctly by tomorrow afternoon I'm going to splint her little foot and see if it helps.
I had something similar happen to one of our ducks that may help you feel better about her chances and/or understand what could be wrong.
Friday before work, I went out to check on everyone before getting in the car to leave. A quick glance said they were fine, but as I walked away, I heard something hitting the side of the pen. It was only once or twice and everyone appeared fine and relaxed so I left.
When I came home, I went to let the ducks and geese out but noticed one duck missing. Then I heard that noise of something hitting the side again - same exact spot as the morning. I had been gone for 10 hours.
Turns out a duck had crawled between their shelter and the side of the pen (which is made of sheet metal) and got stuck. One foot was under her, one behind her, and she was covered in mud from flopping around trying to get out! When I got her out and set her down, she couldn't walk.
Her legs had fallen asleep from sitting in an odd position like that so long. Ten hours, at least. When she stood, she would fall over and flop around to get back up. Finally I just put her next to food and water where she laid down to eat and drink.
It took another three or four hours until she could actually kinda stand, and another day or so until she could walk without a noticeable limp. But now that she has full blood flow in the leg again, she is even running just fine.
Between this recent issue, the way it sounds like your chick got stuck, and the fact that you said she doesn't act like she's in pain, I'm thinking this might be what you're dealing with. Remember, lack of bloodflow to a limb makes it numb, not painful. And with her feet being in the air, I bet the blood definitely drained from those legs!