Clip all the trailing vines with sharp hedge trimmers in stages, starting at the top and pulling sections away as you work your way down. Be gentle as you pull existing vines away from other plants. Small clippers can break up vines that are difficult to unravel.
Remove all the vines down to their woody stems near the ground. Use a hatchet to break up the woody stems down at ground level, and clear the chips away.
Soak the surrounding area thoroughly with a hose to soften the dirt. Shovel the dirt away from around the remaining root and add water to soften it as needed. If you are unable to lift the root out entirely, chop it with the hatchet as each section is exposed until you reach the bottom.
Before refilling the hole, sift the removed dirt for pieces of root or small rootlings you may have missed. Dispose the pieces of root you find.
Bag the removed vines and roots, and dispose of them somewhere other than your yard or mulch pile where they might re-root. Keep a close eye on the former root area for any new growth you might have missed, and pull it up while it is still young.