I ended up reinstalling, and everything is working now.
I took a backup first though, and did a recursive diff of /usr/share/X11 with what I had before and after the reinstall. The only difference was the modified gemini symbol files, that weren't used.
I had multiple layouts defined at one point, but had since removed all but US, and only had the standard shortcuts. I thought I'd removed most of KDE's config too as a test before reinstalling, but didn't keep a copy of the broken config to trawl through. Since I suspect that's where the issue was, I have now and will look carefully if it happens again
Now on to see if I can get more things working!