This plant doesn't seem to spread readily from seed, but perhaps a neighbor is growing it or a piece of rhizome was accidentally introduced when plants were installed, if it was present in the pot of another plant.

Depending on the severity of the infestation, eradication can unfortunately be a tedious and possibly multi-year process because it's such a tenacious perennial. The use of systemic herbicide is the most effective and efficient approach, and you may need to add a surfactant (if the product allows and doesn't already contain one) in order to get the spray to adhere well to the waxy leaves of the weed. Glyphosate is one of the most commonly-used active ingredients in formulations of systemic herbicide. What doesn't enter the plant binds to the soil and becomes largely inactivated and immobile. It is a non-selective ingredient, meaning it can cause damage to any plant it contacts, so apply the herbicide carefully. (You may be able to "paint" or dab it onto the Chameleon Plant leaves instead of spraying, to get a more targeted application.) Surfactants are sold separately alongside herbicides and other pesticides at garden centers since they are commonly used together. The label directions on the products will instruct you how to use them. Even with this tactic, multiple applications will be needed.

If you opt to avoid herbicides, the only practical option is to manually dig them out, getting as many roots and runners as possible. This probably won't be a foolproof one-time solution, so expect to deal with some regrowth from any pieces that were missed. Composting won't reliably kill the pieces in the removed soil, though industrial-scale hot-composting might; you'd have to ask the landfill what they suggest or accept in each waste stream.

Smothering the weeds instead by depriving them of light might work if they are not too intermingled with the desirable plants already. (If they are, smothering could also kill any other plant growing among the Houttuynia patch.) This may take over a year to be successful, though to be fair, so would using a non-selective systemic herbicide. The creeping vigor of this plant risks that they will creep outside of the light-blocking cover during this process, which would allow portions of the colony to survive since the photosynthesis will refuel the roots.

As it is, if any has grown into your yard from across a property line (like under a fence), the weed may persist and return in a future year since you won't be able to kill the original colony. In that case, you could ask a neighbor if they have it in their yard and if they are willing to control it at the same time. Alternatively, you could edge the fence line with garden edging material to help keep it from creeping back in; bury several inches and keep a couple of inches above-grade so you can spot any rhizomes creeping over the barrier, as is done with containing running bamboo, though on a smaller scale in this case. We don't have information about exactly how deep such a barrier would need to be, but a guess would be 6 inches.

Miri