Thank you for your question. A native of Asia, Houttuynia cordata (chameleon plant) is a perennial ground cover that spreads indefinitely and vigorously by rhizomes. As you have discovered, it is very difficult to eradicate. The best method is to remove the plants manually. Dig under the roots and rake the area to get all parts of the plant, being aware that they can root from broken stems and pieces that fall to the ground. Dispose of the removed plants by incinerating them. You will have to repeat this method several times to completely eradicate the plants; you will need to be vigilant.
On large areas where there are no other perennials, you could try soil solarization.
Mow the plants down first then cover the area with plastic or several layers of newspapers or cardboard and add a deep layer of mulch on top. The plants underneath will ultimately die. You could treat with glyphosate or triclopyr herbicides once the area has been mowed down and regrowth has started to show. Several chemical treatments may be necessary, or follow with the solarizing method described above. Chemicals are not the best option, however, as the plant appears to be resistant to herbicides. If you decide to try an herbicide, read and follow the instructions on the bottle carefully,