It is very difficult to kill ivy completely, because the roots are very long and strong and often go under walls or lawns. You could cut it back as far as possible and then dig down as far as you can to apply a herbicide to the roots. Even after years, apparently dead ivy can send out new shoots. It might be easier to cut it all off the wall and then cut back regularly - it really depends where it's coming from.

If the stems are thick enough at the bottom drill some vertical holes into them and pour in some stump killer and cover with a plastic bag to keep the rain out and stop animals being poisoned.

It will take some time and you'll probably get some regroeth from runners but these will be easy to dig out and with persistence you should eradicate the stuff

Or as Melchise said give a a hard tirm and then cut back the new shoots 2-3 times a year which can be done surprisingly quickly.... I use a strimmer!

i have the same problem and ive tried a few things but its going to be elbow grease cut right down to nothing as said and just pull every new shoot up . eventualy it will have to die . i like it but its gone past being nice and growing in the gutter and going up the roof . i cut right threw every stem near ground level about a year ago and it hasnt fased it . ive also used systemic strong weed killer and it didnt fase it .

It must be getting sustenance through the roots that hold it to the wall. I had some on a wall and I used a spade to shear it off the wall. I've allowed it to regrow but only up to about 6 feet in most places and I'm careful to cut new shoots with a stanley knife every 10 weeks in the growing season. That is very quick and easy to do. I can do 20 feet in less than 5 minutes

yes thats what i thaught ie the gutter its grown over . thanx anyway anchorman .