After running into this issue many times and not finding good solutions short of logging out and back in again, I finally found a terminal command that can help figure out what process is using Secure Input:
ioreg -l -w 0 | grep SecureInput
The output of that command will have a bunch of stuff in it, but the important bit is kCGSSessionSecureInputPID=NNN where NNN is the ID of the process(es) that are using Secure Input.
You can then use:
ps auxww | grep NNN
... replacing NNN with the number after kCGSSessionSecureInputPID= from the first command. This will list the process with PID=NNN.
In my case it turned out to be loginwindow, and killing/quitting that process would immediately log me out of my current session, but as the page linked above describes, locking my computer and unlocking it again (via Sleep) resolved the issue a lot more easily than logging out and back in.
It seems likely to me that loginwindow holding onto Secure Input is an Apple bug.
@peternlewis, it might be a good idea to update the wiki page about this to include the terminal commands, and possibly list the processes in the in-app warning in a future KM release.