You should not touch the JKW until the end of august/september or you just sisk spreeding it. It can be got rid off with a bit of patience and knowledge.
Copy and paste from another site on how to deal with it.
irst things first - wait until the end of August up north, early September down south. You need cheap glysophate - may be try an agricultural supplier - a mixing container and a syringe (no needle). You can get the latter at the chemist for about 30p although it can take a bit of explaining. A bit of food dye is useful too and protect yuorselves with gloves.
The mix is glysophate at 5 times the normal concentration (whatever is on the container) with water an a bit of food dye in a container you never want to use again.
Cut the stems above the first node (leaving some stem above to contain the herbicide). Inject 5ml of the mix into the end of each stem, the food dye makes it easier to see which stems you have treated. The syringe is marked, so it is easy to pull up a measured quantity from the container and inject the right amount. The plant draws the herbicide down into the roots as it prepares for winter.
The stems have to be left on the ground (on plastic sheet) to dry out. Burn them later (we go for Bonfire weekend as that caused least objection).
The next spring you will get a small amount of distorted growth, which you can treat with normal strength glysophate - we painted it on because it was in an area where food grew. You do need to wiat until teh growth turn green though - not much use treating it until there is chlorophyll to distrupt.
There will also be a small number of strong shoots, particularly from areas that are between nodes that have died off; the herbicide has just not travelled far enough to kill the root between the two nodes. Leave them to grow strong and then tackle them in the early autumn by injection.
The next year we had very very little growth and most of the nodes had rotted loose and we burned them.
There is one small bit to tackle away from the main area. The pillock there kept pulling up the shoots and consequently there has never been enough growth to effectively treat! So two years after the main area was turned over to cultivation, we are still trying to deal with that little area.
Just a note on the problems of digging it up; the original small problem area became much more widespread the year after the fence was replaced, all along the fenceline. We suspect that the contractors transported tiny bits of the broken root on tools as they moved along the fence line replacing uprights.
And when we had out first bonfire, we added one or two nodes that had broken away during the injection process. The following spring one which was very very charred and partly carbonised started to sprout!
So it is tough.
The major problem we found was that we had very wet Septembers when the regrowth was doing well and we did not have two consecutive dry days to treat the plants. Had those months been dry, I think we would have cleared it completely by the third year.