Where I live, we are lucky enough to have an ivy mite that infests and kills it. For a while I had the only patch of ivy along a back wall, because it was under an apple tree that I used to spray for codling, and the spray killed the mite.
The leaves are too waxy for weedkiller to penetrate, unless you bruise them by walking on them. Misting repeatedly with glyphosate does seem to work, and I think adding WUL helps it penetrate the wax. I have recently cleared a patch around my fence. If it is on a wall, you can cut all the stems about a foot above the ground. If the stems are flexible enough you can dip each stem, for several minutes, into a bottle of glyphosate immediately after cutting while the sap is still running, and it will draw it in and transport it to the roots. If they are too thick, paint concentrate onto the cut stump with a small paintbrush, several times. Trim away the ivy for a foot or so above your cut, so you will spot and remaining uncut stems. You will be able to see any regrowth and treat it again.
I think SBK would also work. It does not kill grass. Glyphosate is inactivated when it gets into the ground, after killing the roots.
Some people like to pull away the climbing ivy at once, I prefer to let it die and go brown, I find it easier to remove when its clinging tentacles are dead and brittle.