Our garden backs onto an old lady's garden, it's huge and she has it divided up with the half that is left derelict backing onto our garden. She has this huge row of evil climbers all along the fence and they are everywhere. I've been attempting to cut it back each year but it's now out of control. Is climbing our lovely tree and attmepting to strangle it, also getting in with our grape vines which are now struggling. It HAS to go. tbh I don't think the lady would mind at all, she is never in that part of that of the garden anyway and the plant doesn't do anything at all other than produce vile strangly stems and leaves. We are organic gardeners so don't really have a clue about weedkillers but we did resort to some Round-up which dH sprayed all over it last week - so far nothing - a couple of leaves look ever so slightly brown but over all it looks just as healthy and is contiuing the campaign to take over the world.
I certainly wouldn't spray weedkiller on it without permission - she can sue you if she wants to. Roundup takes a couple of weeks to work but you probably won't kill it unless you spray the whole plant - not just the bit on your side. Is this it?
If it's climbing and strangling your plants it's not japanese knotweed which grows in clumps of tall stems and does not climb.
It's more likely to be the evil russian vine (aka mile-a-minute plant aka Fallopia baldschuanica) often planted to provide rapid screening but it easily gets out of control. http://www.crocus.co.uk/plants/_/climbers/other-climbers/fallopia-baldschuanica-/itemno.PL 00004749/
Maryz's suggestion is good. We had some at our old house which had eaten an entire shed. We tried to get rid of ours for years in a piecemeal fashion then eventually cut it right back to the ground (it was so well established it had a stump like a small tree) then drilled holes in the stump and injected them with roundup or something similar. It still struggled on for a couple of years after this but we just cut off the few stems it threw out and injected more poison. By year three it was really dead.
We had knotweed and treated it with Roundup Biactive.I used the strongest solution recommended and added wallpaper paste so it stuck to the leaves.In May I painted every leaf on each plant with the solution and did the same in September.It took 3 goes or more to kill some persistent plants,but after 3 years of this,it hasn't come up this year.I know it can lie dormant for years,so am keeping an eye on the area it was in.Hopefully I have killed it off though.I'm hoping yours is something less nasty.