As I've had my KX since 1977, I've gathered quite a bit of information together for '37-'39 1140cc K and KXs. I'm not going to put it all up here, on account of the potential for shady characters to try and make money out of something freely offered.
First up, there is a website for K/KX owners, but it is only available to owners (sensitive databases). You have to email the admin team to gain access: http://www.royal-enfield-1140.com
I will list below the stuff I have, so if you want/need a scanned copy, use the contact me page (including your engine/frame numbers so I can be reasonably confident you are a genuine owner) and I'll get something back to you.
Royal Enfield catalogues 1937, 1938, 1939
Royal Enfield sidecar catalogue 1937
K, KX parts list
K, KX instruction book
Series of restoration articles (3) originally featured in Motor Cycling in 1953
Roadtest of a 1939 model
Large scale drawing of 55 degree cam ring with all necessary info/dimensions for remanufacture.
Scale drawing of Amal handlebar twist dip and horn bakelite controls (Enfield actually used the Bowden version of these, but I did the drawing before I found out the differences). The Bowden version has a much finer serration to the bakelite, consequently most of them are pretty smooth these days! The Amal version retains it's "grippy" quality a bit better.
2 Lucas wiring diagrams to suit panel in tank and instruments in headlight.
That's all I can think of for the moment, if I find any more, I'll keep adding.