Get a herbicide from the garden center, soak the plants and wait for them to die. Do not overspray and poison neighboring plants. The foliage soaks up the chemical and carries it to the roots. String trim the dead stuff to the ground. It takes me 3 weeks or more. By that time the herbicide would have lost it's impact. It doesn't sound 'nice' and doesn't look good either, but it works.

I used to have this issue with my lamium. This year, I carefully deadheaded each stem, and then put some potted impatients and caladiums amoung the plants. The pots help to add color and it pushes the lamium together so it doesn't look leggy. This may seem a bit tedious, but I do the same with my flowers and the results have been great!