Solved it -- dead battery.  NB that I was still able to re-pair it with a completely dead battery (0.4V on the multitester!), and the Windows bluetooth settings reported a battery level of 89%. Moral: don't trust the bluetooth settings battery indicator.
Work gave me a PN557W active pen with my 9310 2-in-1.  It appears to be unsupported, but it worked great for the first three months.  A couple of weeks ago it stopped interacting with the screen.  The top-button still works, and it's still paired. I've tried reinstalling the display driver and the Active Pen control panel, with no joy. Any advice on other (or older) drivers I ought to try?
9310 2-in-1 active pen driver