The general advice I have heard is that basically with a very wet area, trees/ plants generally don't help drain it for you, as they can't deal effectively with dispersing the volume of water needed. I would have thought that even water tolerant trees have a limit as to how much water they can stand and for how long. The following trees apparently have a greater tolerence for wet earth, but may or may not be able to live happily in standing water for prolonged periods.

As far as plants go, many irises love to grow in water and can cope with wet soil at other times.

Evergreen
Chamaecyparis (Chamaecyparis spp.)
Japanese cryptomeria (Cryptomeria japonica)
American holly (Ilex opaca)
Southern magnolia (Magnolia grandiflora)
Austrian pine (Pinus nigra)
Loblolly pine (Pinus taeda)
Arborvitae (Thuja spp.)

Deciduous
Box elder (Acer negundo)
Red maple (Acer rubrum)
Silver maple (Acer saccharinum)
Common alder (Alnus glutinosa)
River birch (Betula nigra)
Weeping willow (Salix babylonica)
Bald cypress (Taxodium distichum)

there are many bog/semi bog plants that would be happy there candelabra primulas, lobelia, ligularia, skunk cabbage, gunnera, lythrum [?] yellow flag iris, Cardamine pratense to name a few.

Should be an ideal spot for white arums.
And any number of members of the willow family.
Digging a natural pond would not only give you a real feature for wildlife and water plants, but would also help drain the surrounding soil of its waterlogging.