My husband has had severe (and I do mean SEVERE) eczema since he was a baby. Our daughters had it as babies but have grown out of it (at least... I hope). So I have a lot of experience with dealing with eczema. This is our tried and true method:
For flare-ups (any time there is a rash), use steroid creams. Steroid creams are the ONLY thing that will calm the inflammation. I've had three dermatologists and an allergy specialist tell me this. Use a hydrocortisone cream as needed for rashes until it's clear, then stop. Steroid creams thin the skin, so use only as needed.
For moisturization, use cetaphil lotion liberally during the day, and CeraVe cream (not lotion) at night. The cream comes in a tub.
For cleansing, at six, she doesn't need soap. Have her soak in water only (no additives or oils). She can do oatmeal baths when she's rashy. Use a light shampoo for her hair, taking care not to get it onto the skin. For the skin, if she's extra dirty, use cetaphil cleanser. It's a non-soap cleanser. Soap, no matter how 'sensitive' it may be, WILL dry the skin. To dry, pat the skin, do not 'rub'.
Use dye-free, fragrance free detergents for the clothes.
If her skin is blotchy and hot to the touch, she probably has an infection, and she needs antibiotic ointment. My husband gets a lot of skin infections, so he has an antibiotic ointment prescribed to him for when it gets infected.
The probiotics are a great idea for prevention.
She needs to take a daily anti-histamine. Children's Zyrtec is great.
Do all of these things diligently, and it works. It has worked for my husband, and his is SEVERE. But you HAVE to be diligent about it.