Fiona,
My sister is just coming out the other side of a similar situation to yours. One of her DDs at 15 was not wanting to go to school, going out and not coming home and had a bad relationship with her father and didn't want to live with the rules at home. She ended up "unofficially" living with the family of her best friend. of course my sister was worried the whole time but my sister and the single dad that my niece was living with had a good relationship and would talk about what was going on all the time. I think you probably have to think about whether the parenting style of the people she will be living with is OK with you. As you need to feel comfortable with that.
While I agree with LS,that she is still a child, you are correct - you can't make her do anything she does not want to do. Some kids just go through this bad stage and you just have to hope they get through it without anything really bad happening. My niece is now back living at home and is turning 18 next month is everything has worked out relatively well. (except that she did not go back to school and has not really worked or got an education ). I am not sure about the board issue. I guess if you give the parent what you would be paying for DSD anyway then that would be fair but I think DSD does need to take some responsibility for money.