At our old house, the City of Atlanta had a kudzu-covered abandoned right of way adjoining ours and 3 neighbors' house lots that they never bothered with. Naturally it spread and kept spreading. I dug up roots until I was tired . . . then we got goats.
The trick is that they keep it eaten back continuously -- the roots don't get any nourishment and they die. Took two seasons to get rid of it completely, but we got rid of it.
Ah, I'll keep that in mind if we ever get kudzu or goats. Did you wait 10 years, though, to see if it's really dead :-)?