Hi Rose7, thick black plastic, cardboard or weed-supressant membrane (which can be expensive) for a year would probably do the job, though ground elder is quite difficult to eradicate. I'm interested to see that you're hoping to turn it into a wild area... ground elder and nettles are pretty wild already. Nettles in particular are fantastic for ladybirds, butterflies and moths, many of which are in decline. And ground elder has flat, umbel-type flowers, which are particularly suited to butterflies and hoverflies. So it might be worth saving time and effort and just leaving your wild patch, as it sounds like it is already a great wildlife habitat.
Kate