Op having read your posts, it seems like you keep coming head to head with DD, understandably.
Remember you cannot exercise a bad diet off, l have found its more like 90% what you eat and how much and 10% exercise to keep weight at a healthy level. Though exercise is important for health.
How about not talking about food for a month, don't give her the opportunity to be angry with you.
plan, buy and cook healthy low fat meals, without any family discussion.
Put it on the table with a variety of veg and when she complains say mmm and change the subject, don't comment on what she does or does not eat.
if she refuses fine, she will not starve, it will do no harm ( most of us could halve what we eat with no harm)
Once she learns that the meal served is what is available she will learn to try new things, keep serving the vegetables in different dishes and not commenting.
If she says she is hungry later, mention fruit or ryvita with honey. Once. Then MMM..
Don't get drawn into discussion or debate.
Have only fruit and healthy options in the cupboards ie ryvitas instead of bread, a supply of crudites to snack on, olives etc be prepared to get rid of the junk or you are wasting your time. Other posters are right sugar is the enemy.
I have one Dc who is a beanpole at 11 but eats modestly, and one Dc who at 10 is prone to being chubby and has a huge appetite.
We have 4 vegetables with every meal,( for 10 yr old l give a very large portion of vegetables) and go for fish/chicken/lean lamb over lasagne or sausages,
to keep him going before before dinner, l only offer crudite or cheese in moderation, he calls this a snack.
We do have dessert but I keep portions quite small, they are not allowed to help themselves to biscuits or dessert items without asking first, but are allowed to snack on fruit, nuts, cheese, olives etc anytime.
We have no cordial or fizzy drinks at home unless there is a party or big event. Only water or fresh apple juice.
This only works as Dh and l live like this, any odd chocolates we may get for late evening treats are occasional and kept out of sight. Halloween spoils, christmas chocolates and party treats are had instead of dessert in reasonable amounts until gone.
Dcs go to parties and family events, out with friends, and have whatever they want, but we all have to learn moderation at home to be healthy and that goes for kids too.
Lastly we don't talk about weight, just about energy giving food,
ie after the Xmas excess l said to DD I wanted to feel more energetic,
( ie lose half a stone!) so wanted more vegetables and salad, and no roast potatoes or yorkshire pudding for a few months as they slowed me down and made me tired.
Ds does say he gets starving even after a large meal so l just say give it half an hour and if you are still hungry have an apple, often he forgets all about it.
Don't make it a volatile subject, be really chilled about it and careful of what you say, often the less said the better.
Remember at the moment you control the choices she has to be healthy or not, don't let it slip out of control as in a few years you will not have any say.
Good luck