I have not had occasion to have to tackle ivy (yet), but have successfully managed to kill off established trees in the past willow, leilandii, and, ash, all close to foundations and drains,
1 cut off and remove top growth leaving a stump.
2 drill a 1/2"-12mm hole vertically into the stump.
3 fill the hole with neat SBK and cover the stump with kitchen foil lid tied on, to keep out the rain.
4 check periodically as the stump absorbes the SBK, top up and replace the lid.
As I see the bark on the stump start to break away I stop topping up, and have not seen any regrowth at all. Plants growing close by don't seem to get affected.
Affect on adjacent tree roots etc, maybe advise from the manufacture of SBK on using this method would be wise, as it is pretty strong stuff.