Dig, then repeat until done.

gypsy_love - I think it would be sensible to start by identifying what you have there. You refer to various 'breeds' of ivy, but most plants commonly called 'ivy' are not true ivy, ie Hedera species, at all. Poison Ivy is not related at all to, for example, English Ivy. Once you know what plants you are dealing with it is easier to decide how to tackle them. The Name That Plant Forum is your friend.

I too agree that it would be best to identify what it is that you are trying to eradicate. See link below to Name That Plant Forum.

Whatever it is, I'd cut and pull as much as possible, and then if possible till the soil really well. Next spring, repeat the pulling process and tilling.

Where are you located? See the Member Pages link at the bottom of any page to update your profile info so it will include your zone and location (see I am 5/6 S INdiana) so that it will automatically be shown whenever you post.

Sue

Here is a link that might be useful: Name That Plant!

If it's largely english ivy forget about id'ing the rest coz the ivy will hide so many others and be the toughest to remove. do let us know if you were successful. ivy is the bane of my yard's existence.