If you still want a lawn in that shady area, ask at a local garden center for assistance in picking out a shade-loving grass seed. You might also try to determine the pH of the soil in that area; moss and grass do not thrive in the same pH, so that alone could be what's encouraging moss and killing grass.
Good plants for shade include
Maidenhair and Venus' hair ferns and many others (but not all); Primroses (for which England is famous);
many (but not all) Violets;
Hellebores (which will appreciate the winter nakedness of the nearby trees);
Dutchman's Breeches;
Trilliums;
Jacks-in-the-pulpit;
Epimediums;
Eranthus;
Dicentra spectabilis (the Asian Bleeding Hearts);
Dwarf Crested Iris;
Wild Blue Phlox;
Hepatica;
Bloodroot;
Claytonia, Dentaria, and Columbine (all of which also love full sun--will grow anywhere but in perpetually damp soil);
Oxalis (which are quite hardy, despite claims otherwise, and do well in dappled sun);
Euonymus americana (Running Strawberry, a woody deciduous sparsely-branched shrub-like plant with mostly green stems, small cream flowers, and bright red berries that birds like);
Dogwood trees and Bunchberry (Cornus canadensis, the creeping, cool-summer, groundcover version of Dogwoods);
Polemonium reptans (blue-flowered Jacob's Ladder, which also can tolerate quite a lot of sun);
all of the Trout lilies;
Sedum ternatum (white-flowered, low, creeping woodland sedum, one of the few sedums that prospers in shade);
Virginia Blue Bells;
Tiarella.
Even in shade, don't forget to mulch!