We have some ivy covering the bricks on our garage, so I read your question with interest. We hack and slash on it 2+ times a year and I can forsee a time when I will wish it gone.
I googled (Bing'd?) 'How to kill ivy?' and several replies said to prepare for lots of heavy work. In Sept/Oct., hack it down as much as possible. Paint the stems as close to the ground as possible with Round-Up. Do this repeatedly at 1-2 week intervals. This will transfer the chemical down to the roots as the plant is trying to store energy/nutrition for winter and 'sap' is moving down toward the roots. Then in spring, pluck off all-every single one- leaf or stem that emerges. This will rob the plant of nutrition derived from next year's photosynthesis hence starving the plant.
Sounds like a lot of work, but that's how it's done. Just spraying with herbicide won't work- the waxy surface of the leaves just shed the chemical. You can crush- or shred all the leaves in your hands or with a string trimmer allowing Round-Up to enter the plant's system rather than run off.
Either way, multiple applications/treatments are required.
Not exactly what I wanted to see, either, but Good Luck!
Jim Ricks