cjhays, I too have a picky eater, although mine is now older than yours (she's 11). We got so worried about her very restricted diet that we actually took her to the doctor and said, please help! Our doctor suggested making sure she eats a very good breakfast with plenty of protein - for my daughter, this means she eats scrambled eggs, toast, and Greek yoghurt with some honey and cinnamon mixed in, every single morning. The dr said if she eats a good breakfast each day, and then is offered healthy foods through the day and chooses some/any of those, she will get through this period of her eating tastes. We also give her a kids' multivitamin daily to make sure she's not missing out on anything vital through her diet. So, my daughter's diet now consists of: Breakfast - scrambled eggs, toast, yoghurt with honey/cinnamon; morning tea - fruit of her choice (she eats apples, strawberries and watermelon); lunch - sandwich on good quality bread, she usually has cream cheese or plain butter, sometimes cucumber; afternoon tea - this is her small treat, she might get a cookie or cupcake, generally whatever I've baked that week, + more fruit or a sandwich; dinner - this is where she really struggles - she eats usually only carbohydrates such as rice, pasta or potatoes, and cucumber. she does not eat meat usually, she doesn't like it. She doesn't eat other vegetables much, sometimes she will eat raw carrot but if you cook a vegetable she will not touch it under any circumstances. She does eat cheese, so cheese and crackers are another snack she will sometimes have.
I guess, try the things others have suggested, like involving her in food choices and preparation, but also make sure in any way you can that she's getting a good range of nutrients. I don't think six is too young to start learning about healthy foods/unhealthy foods, and talking about how some foods are really good for us but other foods are only good in small amounts. Talk about how she's grown since she was a little baby, and how that's because you fed her healthy good foods to help her grow. Help her to see that to grow even bigger and be healthy to run and jump and play and do all sorts of fun things, her body needs good healthy food like your car needs petrol and if it doesn't have enough it will just stop dead. Only maybe don't use the word dead!
I bet there are good books out there about nutrition for kids, maybe your local library would be a good place to look? And you could always take her to a dietitian or your doctor if you are still worried.
I wish you all the best - it's hard work trying to get a picky eater to eat healthily when they resist with all their might!