You have no choice but to use weedkiller - Knotweed is classed as an imported Noxious Weed and attempts should be made to eradicate it wherever it occurs. There is no effective treatment for it other than persistence - usual method is to drip Round up or neat glyphosate inside the cane like stems having cut the tops off, and repeating that every 4 to 6 weeks. Alternatively, there are companies who profess to be able to kill it off or remove it in some way, no idea how much they charge.
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What did you do with the Japanese knotweed you dug out? It spreads from the tiniest piece of root.

It will take at least three years to get rid of the Japanese Knotweed. Our council (Cornwall) has a very useful leaflet about it. You can see a copy online if you go to
http://www.cornwall.gov.uk/default.aspx?page=19740

and select "Guidance for identification and control"
in the centre, a little way down the page.

I see from other sites that stem injection is best carried out in June.

Oh, poor you....you really have my sympathies. It's horrible having to do battle every day in a place which is normally a haven. I have no advice other than an acquaintance claims to have gotten rid of it by constantly digging it up for two or three years. I hope you find a solution.