I just had another thought.
In the pic, it looks like your daughter has long hair pulled back in a ponytail. If your daughter decides she wants to cut her hair shorter, and if she decides she wants at least 10 inches cut off, she might want to consider donating the hair to a charity which uses real hair to make wigs for kids who need them.
Sometimes doing something to help others can be a great way to gain a little self-esteem. A friend of my daughter's did this, and she got so much recognition from others about doing it, it felt so good for her, she is growing her hair out long again to donate it again.
I didn't include the name of the charity that I know of that does this, because I didn't want to turn this into a solicitation for them, but if you haven't heard of them before and you want their name, let me know, and I'll give it to you.
Good luck.