For your 5 year old, maybe you could offer some of the foods she has refused in the past, but prepared differently. BOTH of my kids refuse to eat cooked carrots. (They are 8 and 11 Years old). They have NEVER liked them. But they LOVE them raw. Same thing with bell peppers. They will not eat them cooked (like on a pizza), but they will happily much away if I just cut up raw strips... they especially LOVE the red bell peppers (which I am glad about, b/c they have a TON of phytonutrients!).
I have a friend who's child would only eat green peas if they were FROZEN... like, right out of the bag!
Have you offered her "odd" things, like black olives, pimento stuffed green olives, french fried sweet potato, shredded raw cabbage, steamed broccoli..
The best advice I ever heard (and use) is that even if your kids don't eat/like something... to continue to offer it to them. Put just a small (one small spoon?) portion of the item on their plate at meals. Don't force them to eat it every time, but keep putting it on their plate.
It doesn't always get them to love foods, but they become less averse to trying them out. I always serve something that I know they will eat (not their favs necessarily, but something they WILL eat) as a main dish, a side they WILL eat or really like and usually one item they are less sure about. They may only take one bite... but sometimes we hit on something that one of them loves!
At my kids ages they are required to have one bite/taste of everything. If they don't like it.. they aren't required to have more.