How can tween girls lost weight?

I have a 12 year old daughter who is 5'0 and 140 lbs. She is overweight i know. I'm trying (she is too) to get help her lose weight, but when she started trying the throw up after you eat routine, i knew she is desperate. She wears oversized clothes or anything that hides her body. I need to know how much and what should she eat. We came up with a plan together: she can only eat fruits, vegetables, and unprocessed meats, so I've been making chicken and fish and other meats. Then, I signed her up for Basketball and soccer leagues and she also has her school P.E. If she continues this then will she lose weight? If not alter and change our plan.