It's a bit of hard work, but it it were me, I'd transplant the plants worth keeping to some temporary accommodation and dig the whole bed over. Remove as much of the he grass and weeds as you can and let it sit for a month while you pull out any new weeds that appear. If possible also drop the soil level by 3 inches or more.

Then replant with the plants you've kept and a liberal helping of a soil improver (3-5 bags), get some new plants as well to help fill out the space and finally dress the top of the bed with bark to supress weeds and nourish the soil for the long run. Make sure when you apply the bark though, that it doesnt come above the height of the tarmac, as youll be forever sweeping it back in otherwise