The best way to get rid of persistent weeds is to cover the area with a mulch (carpet, black plastic, for example), preferably during the summer months. After six months or so, lift and dig over... then dig out every bit of weed that rears its head, cover it with weedkiller, burn the area and repeat every three months... It won't get rid of it, but you'll feel like you're doing something.
Seriously, though, mulching is the best.
Dig out as much as you can, but we found plenty of tumbleweed painted onto the leaves was the most effective - good luck! see: www.dgsgardening.btinternet.co.uk/ground-elder.htm