I have a beautiful, near-40 year old red-haired daughter who was a vegetarian from babyhood. I nursed her for a year and a half, but gradually began to offer table food from about 6 or 8 months onward, standard American menu at that point, and from 6-months old, she would spit out scrambled egg, and never would accept meat nor any meat taste. When she was finally old enough to verbalize her thoughts about it, she said it was a texture thing. She did not like meat, eggs, nor milk after weaning. That was unlike any of her siblings who followed. She has remained a vegetarian for all of her life, (but she does eat vegetables, and always did.) Likes cheese too. She is very healthy, strong bones, good immune system... so I guess that dietary approach works for her; maybe it was in her genes, to be a carb nutritional-type. Maybe your daughter is too.
Look into www.mercola.com, and search Nutritional-Typing in his archives, and see what all he has to say about it. According to him, in the human species, there are three main nutritional types, a protein-primarily type, a carb type, and a kind of in-between that eats a bit of each. I'm clearly a protein type, but love vegetables. You might look at what you and your husband test out to be just to become familiar with the concepts, and then consider how that might impact her. Just a thought, worth checking out.
You also might look into juicing and making smoothies. If she turns out to like fruit smoothies, you can add greens and other vegetables into smoothies, where the fruity tastes overpower the veggie-taste, the blender pulverizes the veggies into tiny bits that are barely discernable, and see if she will accept them instead. As well, you may be able to get some whey protein powder and gradually add it into the smoothies, a little bit at a time, where they are barely there, and increasingly there...as a way to get more protein into her while her bones are still growing.