Hi Mike

This is a reportable weed so your council should by rights deal with it for you. There was a bad infestation on the hill down from Fishguard to Goodwick which has disappeared completely, presumably with council treatment. Home treatment is a long job so if you try a DIY job it may take some years. Burning the top growth won't disacourage it much as its very deep rooted. Hard to imagine it was imported by the Victorians as a decorative garden plant!

Our son had some in his garden and was reduced to pouring deep root killer down each individual hollow stem, which worked but if you have a lot could take a very long time.

Legislation in 2014 put the onus on the landowner to prevent the spread of JK. Check whether your buildings insurance will cover the cost. If it doesn't, there's guidance on the RHS website https://www.rhs.org.uk/advice/profile?pid=218

see https://www.gov.uk/prevent-the-spread-of-harmful-invasive-and-non-native-plants to find out what your obligations are.

How ghastly to have either inherited this or had it appear! I hope you manage to get rid. It's a shame that the farmers are not responsible for the infestation of the road-sides of their oil-seed rape! More ghastly stuff, but not of course as difficult to get rid of.

It's interesting that JK didn't really become a problem until the 1980s. We had been happily living with it in the UK for over 100 years, swapping cuttings amongst gardeners without a care. And now we blame the Victorians for introducing it.
And, of course, some people haven't learnt from this and think that it's ok to bring plants home from holidays, send plants to family and friends around the world and so on, with no regard whatsoever for import regulations (and not just importing to the UK - I've several seen posts on GoY concerning obtaining plant material)

The trials of a biological control don't seem to be moving very quickly - but that, of course, introduces another alien species to get out of hand.

Didn't know abut the 2014 legislation - thanks Urbanite.

The enquirer hasn't put their county on their profile which is a pity but this is a link to the Pembrokeshire councils advice.

http://www.pembrokeshire.gov.uk/content.asp?nav=1626,110&id=30502

They are recommending glyphosate but I think you'd have to do it while the plants were fairly small and keep on doing it again as more reappears.

Amendment of Anti-social Behaviour legislation.

Glyphosate is the usual recommendation - don't try to dig it out as the inevitable bits of root that get left will spawn new plants. That's probably where things went wrong in the past - we tend not to put weedkiller on plants just because we want a change - we dig them out but with JK that just led to it growing back.
I think the trick with JK is to leave the roots as undisturbed as possible.
There's a great little map at http://news.bbc.co.uk/1/hi/sci/tech/8555378.stm showing how it has spread since the 19th century.