Hi there. I was battling the wretched stuff yesterday too. In the garden where I grew up, my Dad eventually gave up with the ground elder and grassed over the whole area so it was cut regularly and that did kill it off after a few years. Mine is in a shrubbery and it seems impossible to dig out. I don't use weed killers, but as this is well away from my veg bit I am almost tempted - it is however just a few yards from our well, which is our only water supply, so I don't want to contaminate that.
Apparently you can eat it, but I think that would be a long and slow way to eliminate it