How to help my sister lose Weight?

My little sis (13) wieghs 8.5 stone. All her friends wiegh between 5/6 stone. She is a bit upset about this, her friends dont tease her - but yesterday I caught her trying to make herself sick. I told her I would help her get down to 6 stone if she promised to not make herself sick, and that I would not let her lose ANY more then six stone. We have also been to the docter and he says it's ok as long as we don't go under 6 stone, so how do I do it, she is vegetarion like me - and obviously can't get a gym membership. So what do I do? Any exersie routiens or diet plans would be great!