The ivy on my wall was so thick it had roots and suckers the size of treetrunks. It was totally taking over so I had to have someone cut it down and I sprayed the roots with Roundup, which seems to have killed it off. If I see any sign of new growth I just spray it again. Be sure not to get the Roundup on any of your other plants though as it will kill those as well! Good Luck!

All the parts that root into the wall etc will be able to grow on their own on their roots so you will need to spray all of it. And it may take 2 or 3 applications.

I think you can get purpose-made ivy-killer.

There's only one way I've found - cut the growths at the base and drill into the woody roots and fill the tunnels you make with SBK, cover and leave.