Perennial Weeds

I've been trying to get rid of ground elder since it appeared in a border 2 years ago. I find it difficult to dig up by hand, the stems break easily so it's hard to follow them back to the main root. It's the only thing I use weed killer on. It's definitely weakening, another year and I think it'll be gone, but then there wasn't that much of it in the first place and I caught it early. If it's already well established in a mixed border I can see it being impossible to get rid of.

When I came here it was a thick carpet in all the borders. If I had known what I do know I would have weed killed everything for a couple of years. I covered a few by with sheeting. One I dug up. I've never got on top of it.
My neighbour moved in a few months before me - across the busy road so we didn't meet for a year or so! She was a big fan of gardening and knew what it was like. She didn't plant anything for 2 years, weekilled every few months and dug every bit. 
I wish I had that knowledge then.