I am so sorry you are going through this. And also your daughter, she wouldn't be acting like this if this were enjoyable for her.
I wonder if she feels like she's having something done to her and not being included in her treatment plan? I refused to get braces when I was a teenager because they were talking about a treatment plan that would involved 3-4 years of fixed braces, extractions, elastics, headgear, and a habit thing, etc., that would interfere with my plans to leave the city when I finished high school, and when I asked if my teeth would look perfect after all of that, they said no. And I asked if they planned to do anything about the funky wide circular shape and color of my teeth, and they said, no, this was to get them in straight the way they were currently, and my parents said they weren't going to spend the money to get the color and shape of my teeth corrected, so basically they were talking about I would sound and look like a freak for 4 years, and get my braces off, and still feel bad about my smile.
Much as I don't love going through work at this point in my life, and wish it had been taken care of earlier, and that somebody else had paid for it, I don't wish I had THAT treatment. And honestly, if I had had the proposed treatment, I would have just had a different set of problems. My current orthodontist said, "Well, since you WANT your teeth made smaller, you won't need extractions or fixed braces, we're just going to file and move, file and move, with removable appliances, and then you'll finish with slight spaces and we'll put on veneers. Thank God you never had extractions or we'd have a real mess." I wish my parents had respected the fact that I would indeed be ending my treatment as an adult, and that they had taken me to an orthodontist who had at least been willing to have a conversation consulting with me (WITHOUT my mother in the room to jump in and interrupt me every 5 words) about my cosmetic concerns, my concerns about playing an instrument (I was an all-state clarinetist and was performing with professional adults), that sort of thing.
Anyway, I'm sure there are a lot of things different about your situation, but I know my parents were annoyed and they saw me attitude as being defiant about braces in general, but honestly I did want my teeth fixed, I just wanted to be respected as having a right to some input. I was either 16 or a month or two away from my 16th birthday when all this was going on.