Last fall I dug out as much as I could, covered the area with clear plastic [the heavy dark weed plastic is better, but I already had this], laid down 4-5 inches of newspaper, and covered with wood chips.
When I pulled back the plastic this year, my assessment is - a darn good job.

I still see some white roots, and I still have sprigs of mint coming up here and there, which is what I expected.

I did not think I would totally get rid of the stuff.
But it is reduced about 90%. This fall I'll do it again, with the dark heavy weed cloth.

just to let you know.

Congratulations. I may have to try something like that with the ground IVY that is everywhere.