It takes a real focused effort. Tear out as much rhizome as you can and cover the entire effected area in a smothering layer. I prefer cardboard or newspaper because inorganic root barriers were sent by Satan to destroy us, but it had to be a substantial layer. Hold it down with mulch and/or decent topsoil and watch it like a hawk. Sow native wildflowers the first year, something that will hold the layer together without requiring much maintenance because odds are high you're gonna be back in there tearing it up and finding more ivy rhizome and there's no sense destroying something you love. But you need something there because you're also being assaulted from the air.

Birds spread ivy in their shit. They eat the berries, fly everywhere, and deposit noxious invasives wherever they go. You need aggressive natives to maintain the front line and keep those turd seeds from finding purchase. So you gotta be out there fortnightly to check for little English sprouts as well as hoping the subterranean menace is subdued.

When you have a year with no ivy bring in even more good soil and bury it good, then do whatever you want but never grow complacent.

This strategy applies to most horrible weeds but some cannot be reliably smothered and must be physically removed in their entirety so rent a Bobcat or something and try not to cry.

Let's switch, I try to kill your ivy and you fight my bamboo

Copper nail or a little hole in a thicker wood and inject some bleach in there. It will kill it down to the root.