Poor girl -
I had eczema quite badly as a child so all sympathy. I still get flare mini ups.
From my out of date knowledge it sounds like she needs to hit it hard with steroids to get it down so the skin has a chance to heal - I was always told you have to use a steroid for a couple of weeks after the symptoms have gone to make sure it calms everything deep enough. It sounds like @zaxxon has more up to date knowledge on treatments so ask your Derm about that.
It sounds like she may need some of it to be dressed, either overnight or under her clothes in the day. The Derm should be able to pass you a nurse for that, or the eczema society has good advice. For the mini flare ups I have if I don't get there fast enough I have to dress the bad patches or my trousers / the bed clothes rub and the irritate them.
Cut her nails down to the quick and file off all rough edges if you haven't. Do this every day - and if she absolutely has to encourage her to rub with her finger tips rather than scratch.
If there is anything rough or irritating in her clothes try and swap to soft jersey cotton while things calm down.
it sounds like you are moisturising three times a day which is essential. Aveeno isn't anywhere near moisturising enough for my skin, so I'd consider other options. I love O'keefes, but that is off the shelves while they reformat it. Lipikar Baume (La Roche Posay) is ace but expensive, so I would buy a selection of super moisturising body creams and experiment your way though - Vaseline, Nutrogena, Nivea all make souped up versions for extra dry skin - you just have to see what works for her. (Sali Hughes often has good suggestions - and I agree with her that all the NHS creams are crap.)
I hope it goes down soon. She will need to keep moisturising twice a day for year after it does and at least once a day after that as dried out skin always leads to eczema if you're prone to it.
At the first sign of recurrence, jump on it - it's so much easier to sort it in the early stages.
I've never found what I eat makes any odds, but it's worth trying the basic tests. Gas fire fumes and some makeup causes face flares for me.