K.F.
Part of the problem from my vantage point begins with you and husband not being honest with her. If the doctor has said that she needs to loose weight because she is overweight for her age the truth is that she is probably fat. That isn't or shouldn't be considered a negative thing. Get her active, change her diet to include more fresh fruits and veggies. If she is fit that is much better than being "fat". Everyone has a different body type. Some body types can hold more weight than others.
Loosing an extra 5 pounds isn't the same as being over weight. Her self esteem begins with how you see her and how honest you are with her about how she really is. True beauty comes out from with in but everyone looks at the packaging on the outside. A sad fact but a fact none-the-less.
She is only 6 years old so there is a limit to what she can understand. I would begin by apologising to her for not acknowledging that she is overweight but then not making such a big deal of it. find some other girls that have the same build as her and begin again working on her self esteem. Highlight her strengthes. We all have strengths and weaknesses. Play to her strong points. Get her involved in activities where she can shine or at least have many different kind of experiences. Character building is just as important as weight trimming.