If your garden is overrun by ground elder, you will discover that there are very large quantities of root in the soil. The actual plants are actually surprisingly infrequent in comparison to the rhizome network. Also the root will be well entangled with the root balls of other plants, so getting rid of it all will be quite hard work. And, with a lot of it, you will inevitably leave lots of root fragments behind, which will all regrow. Also, if it has been allowed to flower in the past, there's probably decades worth of seeds to germinate. In fact, realistically, as in my situation, you won't ever be completely rid of it, you'll just reduce it enough that it weeding what is left is no longer a large job, provided you always weed it out th emoment it reappears and never let it start rebuilding its empire. We had to move out one summer a couple of years ago for major rebuilding works, and it now requiring a lot more attention than the previous 5 years.

In your case, a multi-pronged approach may be required. I found glyphosate less than fully effective. So you probably need both a glyphosate and a thorough digging out, the question is in which order. My suggestion, given the current timing, would be to start by digging out all you can easily get at. Then in late summer glyphosate what has grown back (best time to glyphosate as that is when they are taking stuff down to the roots to store for next year). Then next spring dig out everything that reappears. If its still a large problem late next summer, glyphosate again. But it ordepens how easily you can glyphosate without killing everything else. I had the good fortune to be removing it from a bed I was replanting anyway, so nearly all the old plants were being dug out. But the few old established rosebushes I wanted to keep have proved a reservoir of roots that have been hard to eradicate.

Ground elder growing in a lawn is adequately controlled by regular mowing.