To kill the roots you need a systemic herbicide such as round-up or one of the woody shrub killers sprays. To be effective this needs to be taken up by the plant through the leafs.... which you've just cut off. You will now get at least twice as many shoots as you had last years
Apart from digging the roots out, you now need to wait until there is a bit of new growth and paint that with neat herbicide. This may have to be done several times. Obviously, the more active the plant is, the more effective the poison will do its work, so late winter/spring will be the most effective time.