Talk to a nutritionist without bringing your daughter. You may be able to get advice on how to make some changes without it being totally obvious to your daughter. Even though you think you are making healthy food choices, a nutritionist might be able to discover certain things that you don't realize can cause weight gain in your daughter.
Most doctors and nutritionists won't actually recommend a diet for a child in the way that an adult would diet. But they can help you come up with healthy menus and amounts for your daughter and ideas for what else to offer when she's still hungry.
I know you mentioned dance, basketball and soccer. Are these each once a week or does she go more often? What does she do to get exercise on her days off from organized activity? Make sure she is spending a minimum of 60 minutes a day being physically active. This means having fun outside - riding a bike, playing on the playground, running around with friends, etc. Don't think of it as exercise, just think of it as being physically active while playing. I know it can be hard to fit into your schedule, but it is important to try. Encourage her to play active games at recess as well.
Do not convince yourself that it is a lost cause. If you give up, she will give up.
Finally, find out if she is an emotional eater or if she uses eating and food as a coping mechanism. Some people, when they are upset about something or unsure of how to control certain aspects of their life, turn to food. They love to eat and they can control what goes into their bodies. Make sure nothing is bugging her. If you don't think she'll be 100% honest with you, see if there is a school counselor or someone else she could talk to. Just a thought.
Finally, make sure she isn't sneaking junk food somehow without your knowledge. Check to see that she doesn't have a stash of candy in her room. Make sure she isn't trading lunches with her friends or buying junk food at school.
Overall, it sounds like you are doing the right thing and trying really hard to have a healthy child. Don't beat yourself up or take the blame. Don't think you are helpless though. Go to a professional and get the help you both need. Good luck.