would a normal weedkiller eventually get rid of the roots that this has left behind, I have tried to cut the roots as far back as I can but I know that ivy can be stubborn
. . or, for a more sensible answer, the RHS website states:
"Ivy can be killed by severing the stem and treating the stump with a proprietary stump and rootkiller (e.g. Scotts Roundup Tree Stump & Rootkiller, Bayer Tree Stump Killer, Doff Tree Stump & Tough Weedkiller and William Sinclair Deep Root Ultra Tree Stump & Weedkiller)"
http://www.rhs.org.uk/advice/profile?PID=258
We've just cleared our rockery of the bloody stuff, it was everywhere. We hacked away what we could then treated the stumps to a coat of neat Root Out or Round Up, every day I go out and look for any shoots that come up and give them a coat as well. We seem to be winning but I fully expect to keep on finding shoots for the foreseeable future.