Last fall, I got tired of our ivy bed. (Regular ivy, virginia creeper, some other stuff. Thankfully, no poison ivy.) It was constantly overrunning, taking over the bush next to it, running up the wall of our house. My solution was to:
Run the mower deck from the tractor over it once. This just to cut down some of the volume, and disentangle the vines.
Pull the longest vines out as I could.
Attack! With a pitchfork, turn over the entire area, down to a foot deep. This was sufficient to pull out almost all of the roots. As the soil is turned over, it is now easy to pull the roots so no ivy grows back.
Once turned over, amend the soil with some compost. (Might as well improve it as long as I had done the hard work.)
Plant seed, and keep it watered until nature takes over.
All of this took some serious work, but was worth the effort. The ground in that area is still too soft to do more than walk on it softly, but no more ivy climbing the walls of our house. Some time later, there is no ivy remaining, and it has firmed up enough to be able to mow it. While I liked the ivy, it was too close to the house, constantly attacking the wall near it.