Ivy can be a pain in the neck to eradicate. I had to clear a jungle of a garden from ivy and brambles over the last 2 years and have managed to get rid of most of the ivy. Best approach I found is to manually dig out as much of the underground stem system as you can, and keep digging over the seasons as the top growrh re-appears. After 1 year much of it will have gone. Then either keep digging the next season or if it isn't growing around favoured plants use SBK brushwood killer on what remains. By continuing with repeat digging this will gradually weaken it to the point it will eventually die out.