When I was your daughter's age, or maybe a little older, I was very concerned about eating healthy foods, and I went about it all wrong because I didn't know enough about nutrition. Probably didn't help that there was no Internet then. Over the past two years (I'm now 36), I have gradually changed my diet to a nearly vegan one. I've found that I enjoy food more (no need to pick out the gross parts of the meat or consider where it came from), and I feel better and have more energy. I owe most of my new knowledge to the Internet and many, many hours of my own reading.
Although this is a rather drastic example, it might help to read about Kristina of fullyraw.com. She decided in high school that she wanted to eat only raw foods, and her family completely flipped out. However, it's now been almost 10 years and they have all come around -- even her 92 year old grandfather drinks some of her juices and smoothies. She learned what/how to eat without any help from her family and is now a well-adjusted adult who has made a career out of this decision.
I completely agree with two bits of advice given above: 1. Your daughter is still young enough that you should be deciding what she eats, though she should have some choices within a range that is acceptable to both of you. You should not force her to eat meat, and she should not try to force you to let her eat refined sugar all day. 2. Your daughter is old enough to learn about nutrition and do some food preparation herself. Whether kids or adults, people who prepare their own food tend to appreciate it more.
Here are some websites with awesome vegetarian/vegan recipes that have become staples in my house. Maybe your daughter will enjoy looking through them, deciding on something to make, and eating her own creations. Also, if she can see this as a collaboration with you (Hey, let's spend some time together in the kitchen!) rather than a battle with you, it will be easier for both of you. After all, you both have the same big-picture goal: you both want her to be happy and healthy.
ohsheglows.com (Self-taught vegan cook who used to have an eating disorder and is now quite healthy and happy.)
mynewroots.org (Lives in Denmark -- might be some not-so-common ingredients, but many recipes are very doable.)
detoxinista.com (Recently had a child, so there are lots of kid-friendly suggestions and things that are quick/easy for parents to throw together.)
beardandbonnet.com (not all vegetarian, but there is an awesome vegan section under Recipes)
halfbakedharvest.com (might hit home a bit more because the blogger is very young -- I think 19 when she started! -- also not all vegetarian)
rebootwithjoe.com (His films, "Fat, Sick, and Nearly Dead" I and II, on Netflix, would be excellent for your daughter and the whole family to watch. I'm not saying that juicing is the answer, especially for an 11 year old who would probably need to speak with her doctor first, but the idea of rebooting your diet to make it healthier, and understanding that it's a lifestyle rather than a diet, and seeing the amazing changes in people who have done it, is very inspiring.)
richroll.com (A vegan ultra-athlete and father of four (one of whom is, I think, 11!) whose podcast is full of wonderful conversations with all sorts of amazing people, including many who have improved their lives through plant-based eating. His "cookbook and lifestyle guide," The Plant Power Way, is also a great resource. And finally, his article on the "protein myth" is excellent:
http://www.richroll.com/blog/slaying-the-protein-myth/ )
Also, for what it's worth, I was often alone (with my younger brother) in the house after school for a few hours before my parents got home, and we did just fine. We knew that there would be consequences if, for example, an entire box of cookies was gone when Mom got home.
Please keep us posted on how you and your daughter are doing. This could be a great opportunity to improve her health, help her learn the value of researching her own information, and tightening your bond with her over time spent making tasty and nutritious things together.