The bad news is that it will be a combination of hard work and glysophate weedkiller - and it will still take a few years to get totally on top of it.
I was pregnant the first spring we were in our new house - and had to attack a LARGE border (about 15m long and 4m wide!)that the previous owner hadn't touched for 6 years. It was INFESTED - yet it still had lots of good plants in it too.
I soon decided that the best approach was the m hardest work one - ie DIGGING out the ground elder - as I'd never have been able to spot weed such a large area. I was literally sifting the soil with my fingers, trying to get all the roots out - and trying to follw the roots of one plant along underground to the next one.
Fortunately, where I am in Scotland I have both a very rich soil plus it's never too dry (), so if a ground elder root were too entangled in the roots of a plant I wanted to keep, it was easy enough to lift the plant, sort out the roots and then re-plant the "good" plant and know it would survive.
I found I couldn't do that work in garden gloves - I HAD to use bare fingers, even though I was pregnant. I justified it by saying that I'd had cats since I was a baby, they had slept in bed with me all my life and I had never been one for washing my hands after touching them - so I was therefore pretty sure I would by now be immune to toxoplasmosis.
Anyway, ds was healthy, and 4 summers (and a lot more digging) later, I'm now on top of the ground elder (except for a wee patch beside a tree where I can't dig easily - but am about to attack with glysophate!)
I might have won the war on the ground elder, but I'm still battling against the slugs....!