If you guys are being honest with each other, you might want to find out if she is starving herself, or if she is binging and then starving herself. Binging seems like the only way she could be starving herself AND staying heavy. Whatever your daughter might be going through, I know it sucks. It took a caring girlfriend (now wife) to pull me back from anorexia. I really hope she gets this sorted out, because self loathing and bad eating habits are all-consuming and miserable.
If she does have an eating disorder, keep your eye on her, because she is right...a lot of lying goes on.
It is great that you guys are communicating!