how do I kill off well established ivy

Follow the big branches down to the base to find out where it starts. Get a big saw and cut through the main stem. If you then just wait a couple of weeks, the top growth will die back and you can pull the dead branches off more easily than if you tried when they were alive. Once you've done that its easier to see how to dig the main root out! Very satisfying job - good luck!

And if you can't dig out the main root, drill into it and apply a woody root killer, such as SBK or Rootout.