My son has it too. He is allergic to peanut (which he never eats) but didn't test allergic to anything else. You might want to get your daughter a blood test for allergies (not skin test - people with eczema will get a big reaction from the skin test but it doesn't mean anything will happen if they eat it). The blood test can also be misleading though, because it's possible to test positive for something yet never have a reaction to it when it's eaten. In other words, I would only get it done if your insurance will pay for it because it might not help narrow down a possible cause.
My son still gets occasional flare-ups. We wash his face if he gets anything acidic on it like strawberry juice or tomato sauce. The pool does dry him out and he also has trouble if we are outside a lot in the winter. We put vaseline on the dry spots to protect them from the pool and the cold weather. You can put hydrocortisone cream (no higher than 1% strength) on the spots as well. We use a gentle moisturizer too. (We use the plain Aveeno).
The dye-free/perfume-free detergent is a good idea too. It might help to bathe her less often - use lukewarm water and gentle soap (we use Dove for sensitive skin) when you do give a bath. If other family members are using body wash and other soap that's not gentle, she might be reacting to the residue left in the tub. The allergist we saw also said never use Johnson and Johnson baby wash because it's actually very drying.
My son is two now and only got a tiny dry spot this past winter and I noticed a little dry spot since we've been going to the pool. He had a patch on his chin the entire time he was teething so we also tried to keep that dry and protected with vaseline.
Hang in there and keep trying! I know it's frustrating. Our pediatrician is great, but the allergist was able to help us so much more. If it doesn't get better, I'd also consider asking for a referral to an allergist.