Her weight doesn't make you a bad mother, and this is the impression I am getting from you, that you think people may think you are. You need to toss that thought aside and not care what "all these people" say, and simply do the best to your ability to care for your daughter and help her to develop and keep healthy eating and exercise plans.
Possibly the best thing she can do is to eat a healthy diet, set up by you, her, and a nutritionist. But you won't know that until you try it. I would have taken your daughter to meet with the nutritionist when she was 7, because it's always easier for a child to follow a program when they are part of it's implementation and feel they are involved. The nutritionist could have helped (and still can) set up a meal plan around the foods that are good for her and that she enjoys. Serving sizes are as important as what she eats. Along with weight are tests that determine her glucose level, how her kidneys and other organs are working, etc., to determine how healthy she is and what risk factors she could be facing.
You can tell her that the reason the doctor wants her to lose weight is because sometimes being overweight, even as a child, makes us more susceptible to developing certain diseases such as high blood pressure and diabetes (World Diabetes Day is Nov 14 so I'm sure there will be much attention in the media) and that it is always best for us to be as healthy as possible. Diabetes Type 2 has increased tremendously in children as young as your daughter, and it can have life-threatening complications.
To compensate for any low self-esteem your daughter feels you can always have her talk to a therapist, so she is made aware that her feelings have value and that the circumstances in her life do not make her any less perfect. Also remember that this is not about you, as much as you care for your daughter, it breaks your heart, and is a touchy subject for you, you need to not relay this to her, but stay as positive as possible, follow doctor's recommendations and do what's in your daughter's best interests, taking all factors into consideration. {{HUGS}}