As Dr. Phil might say, that horse left the barn a long long time ago.
Clearly her attitude toward food and desserts is way out of line.
I haven't yet read the other answers.
I imagine you'll get some creative and helpful suggestions,
and presume you'll find one or more ideas to help you out of this mess.
If it were me, I'd sit down w/DD and have a big girl meeting.
I would explain about food, about nutrition, about what the body does
with our food, etc. Come to think of it, I'd probably get
some appropriate books from the library with explanations
and charts and pictures of the INSIDES of our bodies.
I would let her know that from now on, in our house,
we will FIRST take care of our nutrition and health needs.
And then and only then, we will consider treats.
That there will be no more negotiating about treats.
And that she will need to cooperate about health and nutrition needs
before she will be allowed to have sweets. Period. End of sentence.
Make up a chart together . . . . proteins, carbs, etc.
Maybe cut out pictures from magazines together.
Lots of different vegetables, lots of different fruits.
Maybe go slow regarding protein . . . .
do some experimentation w/chicken, fish, other mild-flavored items,
to see whether she seems to show a preference . . . .
or, at least, less strident rejection.
This is going to require at least a week, maybe more,
of trial+error but I do believe you and she can accomplish
this together.
It will be helpful, of course, if you (and other adults, older children, if any)
show enthusiasm and pleasure about a wide variety of foods.
Let us know how it goes, please.