Hi and welcome! Sorry you never got a response. You posted a little bit before the Forum went down with a bad virus attack. It was down for most of two weeks. By the time it came back some of the stuff from before got missed.
But maybe no one had a very good answer for you. Ground elder is one of those really nasty ones. If you can't just keep digging and hoeing it out, then I think the answer is just to till up the lawn, rake/ sift out every single piece of rhizome you can get and then reseed.
The more vigorous you can keep your lawn (hard in these heat/ drought times), the better it can compete with the ground elder.
All of the effective methods involve basically starting over. You can solarize the lawn by laying down black plastic in the heat of summer and leaving it for a couple months, to cook everything. You can cover the whole lawn with multiple layers of cardboard/ newspaper (overlapping seams). Wet it down, cover it with several inches of good topsoil and compost and plant new grass seed in to that.
If you don't mind using poisons, you can spray the whole area with glyphosate (active ingredient in Roundup), wait a few weeks for the ground elder (field bindweed) to start coming back (which it will) and then poison again. This will also kill all of the lawn and you will need to re-seed. Since the ground elder is so much more vigorous than anything else, I don't know any ways to eradicate it that don't involve eradicating everything and starting over - except just continuing to dig it out with as much of the root as you can, every time you see it.
I'm fighting a very aggressive new weed called mulberry weed that way - I just walk the garden daily or as often as I can and pull every bit of it I see. It's an on-going process, but I have kept it from taking over.