I finally have it working. I’ve put my findings on the Korg site but am also including them here for the benefit of anyone else who may come this way.
I’d downloaded the latest drivers, which was a dual 64/32 install program for 1.15.r23. That didn’t work no matter what, and I tried it on one 64 bit box and 2 32 bit ones.
I then remembered I had a previous version of the drivers from the Korg PadKontrol (there was midi driver flakiness back then, too). So I uninstalled the current version and installed the older one, 1.13.r6.
Partial joy. Midi Ox showed the Korg under output, but not under input. Rebooted, tried again, same results.
I then went into the config utility. It was set to 64 bit, which is appropriate since this is a 64 bit version of Windows. So, I flipped it to 32 bit and tried again. Started Midi Ox and there’s the Kronos in both input and output.
I then fired up the Kronos Editor and it was able to communicate with the Kronos, sync up and fetch the patches.
The fact that it wasn’t working properly in 64 bit mode and that there’s no 64 bit version of the Vst plugin is probably not a coincidence. Computers and communication have never been a strength of Korg.
In any event, if you arrive here with similar problems, you may find success with a) an older version of the driver and / or b) changing the config from 64 to 32 bit mode.
Hope this helps someone in the future.