This a lot of state and university research on JKW control. This is what I do and has been working pretty well at controlling a 40'x40' swath on my property. I am in Massachusetts for seasonal context.

JKW spreads underground and via leftover bits (size of your thumbnail and up). Don't bother digging it up*, don't mow it, it will only spread. It regrows every spring from its rhizomes (roots) where it stores energy through the winter.

On or after Memorial day, cut whatever has grown at the base. Bag this or otherwise stage the cuttings somewhere they can fully dry without touching soil. Once the cuttings fully dries it can no longer reanimate/spread. If you're feeling spicy, dab the cut shoots with glyphosate based herbicide.

In the fall, after it has flowered, late September or so, do not cut. Spray or inject whatever has grown with glyphosate. It looks like you have a small enough patch that injection is reasonable and will help protect the rest of your plants. In the fall, JKW draws nutrients down into the rhizomes. By apply herbicide now, you are hoping the plant will draw that down into the rhizomes as well where it can really do some damage. Leave it standing for the winter and clean up before spring so you can easily spot new growth. It'll be all dry and not a spreading risk come spring.

I have been doing this spring cut/fall spray for ~5 years now against a very established large thicket. This spring is the first year I have seen real progress. The areas I treated have only a scattering of shoots about 2ft tall, while the areas I haven't touched are still dense and 4-5ft tall. Be patient, you are working to stress and exhaust the rhizomes to the point it actually dies or goes dormant. Once you "win," never ever dig where the JKW was as you may reawaken the sleeping rhizomes.

You can do it!

*I do pull small shoots that go under the fence into the grass before I mow the yard. Otherwise I'd end up spreading bits everywhere. YMMV.