Hi there,
The trick is, if you want to poison it with a weedkiller is firstly cut the vines off say....a foot or so from the ground then treat the cut ends of the stems with a weedkiller containing 'Glyphosate' such as 'Roundup'
Cover the treated ends with a polythean bag to stop it getting washed away, so tie it on with some twine or wire.
Other weedkillers from your local garden centre that may also be effective are 'Deeproot' , 'Rootout', Weedol'
Ivy leaves are very glossy, so treating the leaves will be like water off a duck's back and will be a waist of time, this is why treating the stems is the prefered method but you must give it time for the chemical to work its way systemicly down to the roots. The part of the ivy above where you have cut can be pulled away to remove it.