I found the only way was to dig and riddle/sieve the soil, twice. I had a 30x 6ft bed riddled with it. I lifted the plants I wanted to keep and cleaned off their roots very carefully and planted them in larger pots than you'd buy them in, to give them a chance to grow on.
I didn't plant anything back in the bed for 6 months. then any ground elder 'pips' I'd missed showed growth and they were easy to remove. It has been successful. hard work but it was strangely satisfying.

There is no quick, magic answer to this one I'm afraid. Essentially, I agree with Seaburngirl, I did a similar thing - removed the planting, cleaning the root balls of any trace, planted them elsewhere for a while. From there, we differ slightly - I watered on glyphosate, waited a fortnight, then treated with glyphosate again, waited a week, and then dug it all out. The glyphosate might appear to kill it, it doesn't, but when you dig it out, the fine, hair like roots have been killed, leaving behind the thicker, more visible roots which are much easier to extract.

I also used a combination of digging out and then weedkiller on the reappearing young shoots. Then by chance I planted perennial Geranium macrorrhizum Ingwersens variety and as it clumped up it stopped the young shoots coming back. I repeated that in other areas of the garden with ground elder and other macrorrhizum varieties and it worked. You have to keep a careful watch in the early stages ( and later too ) but now I am virtually free from ground elder and of course, have too many areas of geraniums but after a while they can be thinned out and replaced by other plants