What kinds of weeds and grasses? Shallow rooted annuals? Or rhizomatous perennials?

I afraid that it means digging it out completely, (which is what I have just done) and starting all over again. That is what I recommend anyway, put plenty of fresh compost in and re-plant your plants and take cuttings etc, its amazing the difference it makes and gives everything a new, fresh start.

Regular grass from the lawn and rhizomatous perennials as well. Digging up the whole garden is not an option. Way too labour intensive right now as this garden is about 30 feet long and 6 feet wide.

Just start at one end and do it as and when you can, trying to maintain the part you have done already as you progress. If it is possible to protect the perennials with card or paper or something and spray in between them with glyphosate it should slow the weeds down until you can get round to doing them properly, but any spray at all landing on a plant will damage it so do it on a very still day and be very careful about protecting what you want to keep. You may find that you have to dig up the perennials and split them in order to untangle the weeds, but you will then have several new plants to put back again. Don't do this until the autumn though. There is unfortunately no other way!

Edging the lawn will make it look a lot better in the meantime and also prevent more grass spreading into the bed.

That is what I have done Stera.g and my garden is about 80' x 24' approx. Hard work but no choice!

Wow Grandmage, you must have worked your socks off!

Sure did, and it is still 'work in progress' !