right you can kill it chemically but it takes ages
whenever there is new green growth spray it with a glyphosate based weed killer (remember this kills everything so don't splash on stuff you care about) keep repeating every few weeks for months and months and eventually it will die..
remember it is only absorbed through fresh new growth!!!
digging it out as much as possible speeds the process up a lot but usually you need to get chemical with it to finish it off!!!