Depending on what state you're in, there may be a state funded school that you live at. I went to the Indiana academy my junior and senior year of high school, and it's basically set up to mimic college but with more structure. When I went it was 100% free to go, now there's a small cost each year. I was a lot like your daughter when I was young, and I was afraid to advocate for myself. Going to a new school with new people really helped me open my shell and be more outspoken, etc. in all honesty it sounds like she's unhappy at her current school but doesn't want to tell you why. The really great thing about being away from home during high school is that she will get practice interacting with a roommate, and she will have to learn to advocate for herself- and the residence staff will be there to help her do so.

I truly believe that going away to school before going to college is one of the smartest things anyone can do. I made friendships there that I will have for the rest of my life. (14 years and counting).

I only read the top comment, but I personally would consider it if I were you and I could afford it.