I feel your pain. I dealt with this pesky vine for some time. It is tough to kill, but you can do it. What I did was to cut each vine and paint on or spray on a product called "Tordon". It's a brush/tree killer. You need to get the chemical on as soon as you cut the vine so it gets into the flesh of the vine before it scars over. It will travel down to the root system to kill it. It's tedious doing it for every stem of the vine, but not that bad. I had a lot of vines, so I just poured the Tordon in a spray bottle and sprayed it on the tip of each freshly cut vine. Here is the chemical--it is blue so you can see where you have sprayed it.
https://www.walmart.com/ip/TORDON-RT...SABEgJg7vD_BwE