If grass is growing badly there, then it's no place for that sort of grass.

If the weed you have is the sort that can be mowed - and contained from spreading you could resign yourself to making that your ground cover.

You could try smothering the area thickly with untreated sawdust which tends to bring creeping stems to the surface and makes it quite easy to pull.

You could lure one of your children over and ask them to remove the soil and weed, and trailer it all to the tip/dump before returning with some clean topsoil, or lay some building sand and pavers so you have an extra dry play area for grandchildren, dogs, the favourite lounger - and pots...

You could actually cover the area with precious compost to improve the soil, resow with a locally tough grass that will take the condition you have on that west side and let the weed become a minor nuisance or even die out.

You could cover the area thickly with chippings from the tree maintenance people and zap any of your least favorite weed when it shows above ground. If you cover it with a bucket with a brick on top the dogs should be safe, and do it between visits from the small folk. I notice someone recommended that you ask for material that doesn't contain brambles or some of the 'instant sprout' trees such as willow. (Another place to stash pots...:-0)

Hope that sparks some ideas for you.

Try to see if your County Agent could pay you visit. He could identify the weed, and you could then try to get rid of it by following his recommendations.

If that does not work out, get one of your better lawn servicing agents to visit you and give you a recomendation on the type of weed, and how to eradicate it. Ask him first if there will be a charge.