I was going to put my 2 cents in on your post but haven't had the time til now, sorry it's late!
I am probably the youngest person in this thread, at 20. I started gaining weight when I was 14/15 and have never lost a pound since then (until March this year of course)
What other people have said is right... that age is really difficult to trust with diet related things. I know your daughter is likely mature and intelligent, as you seem to be, but when I was that age everyone thought the same of me and I can tell you, I was not. Just before I got really big in the first place, I went on a restrictive diet. I went way overboard and got way obsessed. My parents were really concerned but I didn't think there was a problem. Such irony that I wasn't even fat then!
Remember that teens have so many things in their lives that we don't. Other teenagers can be evil and mean, the media has so much more power over teenagers, and the internet also hosts a huge amount of questionable diet/appearance related content. Even as an intelligent teenager, it's really REALLY hard to not let that affect you. Everyone always told me how smart/mature I was, sometimes I felt like a nana for my age. All adults commented on my maturity and intelligence at that age. But when I look back now and remember some of the beliefs I secretly held, some of the things I thought about, and some of my goals - they were by no means the choices I'd make as an educated adult.
Anyone who has weight issues will surely tell you, they wish they just learnt to eat right young, and never went on a diet, right? I would want that for my daughter. Don't start her on a cycle of diet and failure so young when honestly, as everyone else has said, eating paleo or primal and low carb will drop her weight really fast anyway AND teach her skills for a lifetime of being healthy without diet cycles. Teach her this one thing right and she'll have the rest of her life to thank you. She'll be a healthy, educated young woman by the time she's 18 and that's honestly the best start you can give her for college or whereever she goes. God I wish I had started uni at a healthy weight!
What you do in the end is of course up to you and it's great that you've decided to do something for her. Unfortunately when I gained all my weight I was not in a place where my parents could intervene in any way haha. Good luck to you and her