Using an herbicide that will only kill top growth will not work, since the extensive root system of houttunyia is your biggest problem, and that does not eliminate the problem. You must use a "systemic" herbicide that is absorbed into the system of the plant and goes down and kills the roots. The most common systemic herbicide ingredients you should look for on an herbicide label are: glyphosate and triclopyr.
Glyphosate is a total vegetation killer, so be careful how you spray. Do not let the spray contact your desirable plants.
Timing is very important with these most invasive of plants. Besides "eternal vigilance", i.e. removing every bit of above ground growth you can, you will want to use glyphosate (possibly triclopyr), keeping in mind that there are two best times to spray:
1. when foliage has been removed and new foliage is coming in. Zap the tender new foliage.
2. Best of all, spray in the fall--2 applications. In the fall, plants are drawing carbohydrates down into their root systems for winter, to be used to produce new leaves next spring. They will pull a herbicide down into their roots the most efficiently at this time of year.
So, in early fall (late August-early September) spray once.
Wait about 3 weeks, and spray again.
And spread the word that this plant should never be sold or shared. It's a shame that anyone has to struggle with this nightmare of a plant.