Four years ago, I did battle with Japanese Knot Weed in the back garden of a house we rented in Cambridge. I tried to kill it for two years by digging it up and applying weed killer. It was very resilient, but I was winning the battle before we left that house.
This was a small area looked after by a pretty determined individual (me), I can't imagine what it would take to get rid of it from the the huge areas it occupies such as the valleys around Cardiff.