What kind of bit/bridle set-up are you using on your horse for these lessons?
I just ask because I like to keep my mare/lesson pony in the absolute gentlest thing she'll respond to for the first few lessons, while the kids figure out reins. In her case, it's one of those "riding halters" - the rope kind with rings at the nose. Her response time is a littttle "gross" in it compared to a bit but, imo, for new riders, it's ok if the horse is a little sticky. In the beginning, while the horse's patience is being tested to the max, horse comfort is #1.
Later, once the child becomes more proficient, I switch them to a bit but we go back to bitless for new things, like trotting/cantering, where the horse's mouth might get accidentally popped. Eventually they do everything with a bit but to start, everybody goes bitless and as gentle as possible.
The other thing I've found, with kids who always seem to end up with their hands up near their heads - either the reins themselves are too long for the child/horse combo [I keep a variety of rein lengths and sizes to accommodate for different kids - the first reins all "my" kids use have knots in them at the proper place to hold for plain riding and knots at where they should hold for turning] or the kid isn't being challenged enough. If I keep them engaged, via games and other exercises, I can usually get an entire lesson where hands stay down+are correctly placed.
In your kid's case, I might stick her on the lunge line and really drill her hands for a lesson - if she wants to ride on her own, she needs to do x, x, and x while keeping her hands correctly placed [like change directions on the lunge line, stop the horse, change directions again, whatever "gets" her]. Basically classical conditioning. She goes back on the line if she starts getting her hands up too much.
You don't want to be mean or anything about it but correct hand placement = riding by herself, incorrect hand placement = lunge line. She's in complete control of the "reward" [off line] and in complete control of the "less fun".
You can make the lunge line fun by having her stand in her stirrups while turning/cuing the horse [I find this REALLY helps a lot of kids -I like to start all my lessons with at least 5 minutes of standing in the stirrups for 10 steps, 15 steps, etc], play red light/green light - it's not "punishment", it's just a more controlled environment for you to get her attention.
And try just changing the reins you're using. I've noticed some dramatic changes in my kids just from a simple change of rein length/size. One girl in particular does just fine with a pair of 9ft roping reins that are about 2/3in in diameter, but switch her to 7ft barrel reins that are 1/3in and watch her suddenly start to ride REALLY WELL!
The first time I just accidentally switched it up and I couldn't hardly believe what I was seeing! Haha