I don't know if you're opposed to herbicide, but triclopyr will kill it. You can get triclopyr salt (water based) and apply it to the freshly cut stump surface (within moments of cutting), or triclopyr ester (oil based) and apply it to the outside of the plant close to the base, no cutting required. Both of these will kill the root. Otherwise just keep cutting and eventually you will exhaust the root.

Not sure about elsewhere, but in the USA you can typically buy the water-based triclopyr salt in a small bottle with a brush attached to the cap. This is in pretty much any garden store. Even though you have to cut the plant first I think this is the best form for just a few plants.