Actually, I managed to get it working again late last night (wow- I say it that way, and it sounds like I actually did something).
In any case, I'm using the same Kindle firmware I was previously, though I did apply some version updates to Calibre since the last time I connected the Kindle.
In reading through some other threads, I noted that Calibre seems to somehow work with/through iTunes. That made me wonder if plugging in my iPod just before the Kindle might have had any effect. So, I ejected my iPod, closed Calibre and iTunes and then restarted both. Once I connected the Kindle, everything seemed to be working. So, either having the iPod connected throws off the rest of the system, or it was all simply coincidence.