Good golly, that is one tenacious plant that should probably never be let loose in a garden. If you don't want to use Round-Up, which will work, you can use Tordon RTU. Even though it is suggested for use on woody plants, I know it works well for houttuynia because I used it to rid my rock garden of it some time ago. Cut stems at a diagonal, wait for them to stop weeping, paint just the cut surface with the product. I used a water-color paint brush. You'll be using such a small amount via that method you needn't be concerned about nearby plants.
If you're willing to pull it up as soon as you see foliage, with as much of the rhizomes (that reach several feet in length) as you can get, you'll eventually be rid of it, but it will be work.
Al
