Oh dear, what a nightmare. My experience of ivy says that, spray with whatever you like, it's like the terminator and will be back. Perhaps napalm works, but that'd kill everything else as well. I'm afraid the only answer is to dig the stuff out properly, and treat any large, woody roots you can't get out with SBK - which isn't good for other plants, so be careful when applying it. You need to make cuts or drill into the wood and fill the cuts or holes with SBK.

At least the roots are not persistent like say ground elder or couch grass. Once you have pulled one out, anything left will not regrow. We get 1,000s of bird sown seedlings every year.

Owdboggy, I wish that were true of established plants. I fought with one on my north wall for years. I even dug the roots out to a foot deep, and it still came back up. It took three years of consistent spraying with brush killer to stop it from trying to eat the house! What helped a lot was using a surfactant with the herbicide.

You must have been very unlucky then. All the Ivy from seedlings to ones growing up a 60 feet tall Pear tree have disappeared when the majority of the roots were removed. Now I agree that any piece of stem left will have millions of dormant buds and regrow if merely cut down, so I suppose that a buried stem would regrow.
Agree about the use og a surfactant, that does help a lot. What did you use? We just added liquid soap to Roundup.

I used Hi Yield Spreader Sticker. I've found out that soap sometimes degrades some of the herbicides, though the difference isn't always noticable. I don't think that I left any stems behind, though in a massive digging project you never know. Suckers came up all along the wall, from roots under the house...I thought.

Ah, right, then they probably were more like stems than roots.

Hmm...I thought they were roots, but I guess that it could be hard to tell the difference.