pull them up dont cut them as that will encourage more to grow. Try putting weedkiller onto the damaged stems too. there i obvioulsy plenty of original root left in the ground. hope this helps. you may well get other suggestions too.

You need to treat any large/medium/small roots still left in the ground, however far they spread, by first exposing them, then drilling into them with a wood bit to make little tunnels and fill the tunnels with SBK (don't spill it anywhere, don't get it on your skin). Cover with something and leave, re-treat if more appears.