The OP should say nothing to the little girl. She knows she's overweight. She's probably teased to no end at school. All she wants is to be loved and accepted exactly as she is. Pointing out that she is obese will only hurt her feelings and smash any self esteem she has.
The best thing she can do is to love her just the way she is.
I would agree completely if she were 25, but the girl is 12. She's not nearly fully-formed as a person yet, and that is exactly when role models are a great idea--not to bash her or to replace parents, but to model a different set of behaviors from the ones she typically sees. She deserves to know she has options.
I like the mall idea, and having her help cook, although I agree with the poster who felt that asking her to help you fulfill fictional doctor's orders will likely backfire. Maybe just something like, "I really feel like getting out. Want to come to the mall/local park/windowshop downtown with me?"
I live in NYC, and I used to have a friend who called one night to ask if I wanted to take a walk. I soon found out that she felt like walking fairly briskly up and down about 20 city blocks, zigzagging at the end of each column so that she didn't cover the same ground. She was afraid I would take it wrong if she described it as exercise over the phone (I was distinctly un-thin), but she didn't think she would go if she was alone, and wanted company. Because we were close and she made it clear that this was a thing she
did and that we could stop for coffee instead if I wasn't in, it never occurred to me to doubt her motives, and it ended up being a regular thing.
If nothing else, seeing that mild exercise can be a voluntary thing, rather than a chore, might help.