With most ivy, you need to get out the roots. If not, it comes back.
First, I'd cut out what I can right down to groundlevel. Then, I'd roto-till the soil. THEN, go and buy a Kill-All vegetation killer. The Big Box stores sell it. It kills EVERYTHING in the soil, including good bugs and such. But, BEFORE you leave the store, I'd read the Instructions. There are some things you should know BEFORE you leave the store. In short, you usually need 3 days to wait after applying the stuff. And pets need to keep away. AND, you need to refertilize, and such. (Read the container so you can plan ahead to buy stuff like fertilizer, and such.)
As far as preventative measures, you should have the problem down to a much more manageable level. You know, like a few times a year, you may spot ivy growing in this area. Just remove it like you would Dandylions, Crab Grass, and the like. It's just maintenance.