Hi, C.! I am a pianist and was a piano teacher, and I think it's absolutely wonderful that you want to supervise your daughter's practice.
Does your daughter's teacher keep a notebook of what she is supposed to do each week? How much is she required to practice a day?
When I was teaching, for kids her age, I usually required just 30 minutes a day. I think instead of checking up on her every move when she practices, set a timer for 30 minutes, and have her fill in a practice chart. It's good to be around when she's practicing to help her with questions she might have or encouragement if you think she's getting discouraged. But if you hover, she will get frustrated and won't have the chance to see that she is responsible for her results. Let her be accountable to her teacher; she is old enough at this point to see how her hard work will reap rewards. Conversely, when she doesn't practice, she'll be stuck on the same pieces week after week, and that is no fun for anybody!
Here's an idea for helping structure her practice time. If she has 3 pieces to work on and 30 minutes to practice, have her spend 10 minutes on each piece. And then let her work the piece through on her own. Eventually she'll realize that some pieces need less work and others need more, and perhaps she'll spend one entire practice session on one piece and pick up the other pieces the next day. Or maybe she'll be so excited that she'll keep practicing beyond her 30 minutes one day!
Also, if she is stuck on a passage, there are some things she can try to perfect the difficult passage. I used a 3x practice rule, where I had to practice a measure until I was able to get it three times in a row perfectly before moving on in the piece. Or I'd start at the end of the piece and learn the last line first, then the second to the last line, then the third and so on. So when I would play through the piece, I would be playing toward the easiest part, instead of always having the hardest part at the end. Another thing to do for a tricky passage is to play the measure with different rhythms: for example, if there are 4 eighth notes in a row, she can play long/short/short/short; short/long/short/short; short/short/long/short; etc. You get the idea. :) Give her tools like this to work through the hard parts, and be available for her while she tries them out. Then let her use her new skills on her own during her practice time. And then only step in when she really wants or needs you too.
Does she have any goals to work toward in her piano? Are there recitals or piano auditions for which she can get her pieces ready? Does she have an outlet to perform at all for family/friends?
It's VERY unusual for a child to be self-motivated when they are taking lessons, and if the parent wants them to continue (until the child is hooked on their own), he or she does need to be involved in structuring practice time, which you guys are doing. Just give her a little more freedom AS she is practicing and let HER be more accountable to her teacher and herself.
Sorry this is SOOOOO long! If she really wants to quit, maybe tell her if she is really unhappy after taking lessons for 3 years, then she can quit (or two years--whatever you are comfortable with). At that point she'll have learned a LOT, and will be able to take that knowledge with her through out her life.
Good luck with this all! Maybe you all can get over the hump this year, and she'll keep on going. And as a teacher, thank you for being wanting to be involved!! :)
God bless!