We use a product called Roundup a lot here, or Glyphosate. It will kill a lot of stuff. Spray it on and in about three hours you will know if it's gonna work for you as the plant will usually begin to wither. It's absorbed into the plant then taken into the roots for a complete kill. Don't get the stuff on you or any of your critters because it's a class 2A carcinogenic but only probably carcinogenic with no long term effects as microbes in the soil break it down after about 50 days or so. Still it's probably better to play it safe with your pets and such until it gets a chance to dry.
If you were to try to dig up all the kudzu in this county it would be a lifelong endeavor as the stuff has spread all over the Southeastern US now thanks to the Department of Highways for the State of Georgia who turned the stuff loose here.
