i've got the same problem, only i have neighbours on both sides who don't do gardening, and ground elder is endemic to our area, so getting rid of it for good is probably never going to be an option. (i've considered asking both sides if they want help, but have a feeling that once i do, i'll be expected to keep on doing it and i really can't afford to be gardening for free for all and sundry!)
when i moved in i had a whole garden covered in the horrid stuff, i've now managed to push it back to the 'wild' area at the back of my garden and use a combination of rooting it out and spraying. where it comes up in the lawn i've used sbk, as it doesn't seem to damage the grass, but keeps on top of the ground elder, borage, nettles and brambles.
on the worst area by the house i gave up trying to keep it clear and instead used weed membrane and gravels to create a patio type area where i keep my potted plants. occasionally i get a bit of ground elder popping it's leaves up but a bit of glyphosate painted on seems to do the trick and i've even felt brave enough to start planting up through holes cut into the membrane.