Getting rid of Japanese knotweed [aka Japanese Bamboo] is do-able; doing so requires persistence. I got rid of a patch somewhat bigger than yours by digging out the plants as best I could and then by assiduously cutting back new growth as soon as it appeared. [This starves the plant for light and interrupts photosynthesis.] After the initial digging out of individual plants, the maintenanceÂclipping new growthÂwas pretty easy. Even mowing works; it just takes near-daily diligence. The patch never recovered and within two years was 100% knotweed-free, and has remained so for the past 15 years.

Herbicides work, too, but they will kill the juniper as well and will not completely kill the knotweed's underground root system, so you're still left with clipping. If you do choose the chemical route, spray the knotweed when it is in flower (late summer). But try the non-chemical method; it repays efforts with a lot of personal gratification.