REALLY gorgeous birches Afban! You've got enough headroom on the surrounding walls to add some more compost and other organic matter. If the problem is extreme dryness you could incorporate the water retaining gel that is sold for hanging baskets. I find it's useful to hold moisture in very dry soils.

You could try planting variegated vinca's, epimediums, hellebores, Campanula poscharskyarna. Bulbs that flower early before the trees come out should be OK & cyclamen do well under trees. I've found that Pachysandra struggles under trees, just as you've discovered.

Personally I would think it highly unlikely you could grow anything under those Birches! They look wonderful but will be taking all the nutriment from the soil. I honestly doubt that water retaining crystals will help. How about some containers with plants growing in?

I've just remembered that in a previous-garden-but-two we had a large silver birch and under it grew crocuses and trailing nepeta. :-) Proof that 'something' can survive under there! Just add the compost / organic stuff you were going to and rake in some slow release fertiliser every year.

couple of things come to mind. A trickle water hose system under the surface of the soil, obviouse drawbacks are intitial installation/cost of materials & water-bills if on a meter etc but you can turn it on & off as & when you feel it necessary. Then you can plant any & all of the shade lovers which you can then grow with confidence. Personally this is over the top but it's a solution.
Secondly & one that I would be more inclined to do would be the good old compost addition as has been suggested & I would chuck a REAL HEAP of it on top of the whole plot, fork it in as best you can & I know how fiddly it will be with those tree roots then bark chippings as an extra mulch & ornamental finish.
Plant foxgloves & see how they do, if happy they will self seed & you have them year on year, white is a good colour in the shade but tough purples equally excellent. Hardy cyclamen, Coum & Hederafolia are both superb bulbs both for leaf & flower, one flowers in Spring the other in Autumn. Smaller daff bulbs as opposed to the tall varieties.
I mention these as I have them all growing in similar conditions as yours & it has worked really well. Good luck.

I've answered this in your other question/post.
In my very dry tree-shaded areas i grow pachysandra, geraniums, huecheras, tiarellas, phlox, ophiopogon .... i could go on and on.
If you plant as deeply as you're able and incorporate a bit of manure this will help.
Trying to plant larger/mature items will be more difficult than planting younger, immature ones - 'they' will get their roots in easier.

The problem with building up the soil and keeping it moist is that the roots of the birch will quickly grow upwards and fill this space also. You could either keep the existing ground irregated or, maybe, use containers for other plants - again, regularly watered as the leaf canopy of the trees will stop rain falling onto the containers.
Bizzyb's suggestion of small, early flowering bulbs is also good as they will flower before the tree leaves block out the sunlight and they prefer a dry summer dormancy.

i agree bt piling loads of well rotted compost on ,we have a woodland area 2 and plated ferns bulbs etc. things that seed them selfs r gud.as in wood. the worms will draw down the soil and breack it up

Thanks to everyone for this advice. I'm gradually removing some plants ready to get started with topping up the soil level etc and we're looking forward to replanting the area.