Hi B., I can empathize with you. There are some really good ideas in the previous responses, especially with adding the Flora.
My son had a similar problem shortly after going off of breast milk. As it turned out, he was allergic to wheat and milk products so we put him on Soy milk and goats milk. Even now at 16 he occasionally has problems which seem to coincide with his growth spurts and foods he's allergic to but he sneaks into his diet. The body is using more nutrition for growth, so the intestines seem to know when to hold onto the waste and pull more nutrients out of it. About 10 years ago I put him on a Blood Type diet (Eat Right 4 Your Blood Type) and many of his digestive and behavioral problems went away!
Dehydration also seems to contribute. It's hot outside and cooler inside. The air conditioner will dry out the air in the house thus pulling moisture off of the skin and leaving the bowels dry. Try to increase her juice/ water content.
As far as vegetables go, the carrots and squash are high in natural sugars and will help with softening the stools. She seems to know what her body needs! I found increasing natural sugars in my sons diet- then and now- seems to help keep him regular. The sugars that worked the best were brown rice, squash, apple, grapes(now) and grape juice. I stayed away from processed sugars since diabetes ran in my family. For kids that have allergies to rice, it usually causes extreme diarrhea or constipation.
I also massaged his belly a couple of times a day. I'd start beteen the pubic bone and his left hip and gently press- like your pressing dough into a pie pan- in a counter clockwise fashion. This help release the 'gate' and soften the stool in the colon. After doing that once or twice, gently rub the bellie in a clockwise fashion to stimulate gentle intestinal contractions. He's usually have a BM w/in 30 of doing this which helped keep him from becoming impacted in the first place. If you explain to her that big people often do this on thier own w/o thinking about it, she might be more receptive. I started this with my son when he was 6 months old, so by 2 it was a routine we did every morning and evening and provided mommy and me time when my daughter came along.
I love prune juice, but my son hated it. When I had an impaction problem after having surgery, the nurse gave me a 'cocktail' of prune juice and milk of magnesia. After 2 drinks, I was cleaned out and flowing free! lol. It was quite embarrasing for me as an adult, so I can imagine how your daughter feels.
Hang in there. You'll find the magic combination that works with her.