I saw one of these nanny type programmes once where they dealt with food phobias in toddlers.
What they did was to choose a piece of food that they thought would be most acceptable to the child - maybe something they have tried before and eaten - even if it is not necessarily a healthy choice, maybe like quavers or cheese or even chocolate.
They then encourage the child to touch the food - if she does, they go mental, clapping and smiling at her, if she doesn't , they just take it away and try again next time. I think they were also rewarding with a piece of food they liked or a sticker or something as well if they were old enough to appreciate a reward chart.
Once they were able to touch the food, they would be encouraged to kiss it or touch it to their lips. Again massive feedback if they did it.
Next stage was to put it in their mouth - even if they then took it out again.
Next would be biting it, again not necessarily swallowing it.
Next chewing and lastly eating - if they progress faster, you can obviously miss out stages.
Once they eat one food, you progress to the next and repeat the whole process.
It takes a long long time and you need a lot of patience, (I think it took a week with one child to get her to the point of kissing a grape!) but it seemed to work eventually. I guess with your little one, you could give her some of her milk, but reward her with a little bit more if she managed to touch the food etc.- that way you can still give her the milk she needs but it becomes a bargaining tool as well.
What an awful situation to be in, I hope that something works for you.