OK, I completely solved the problems with Kmplayer, and half-way with Kaffeine.
This is what I did.
I deleted the Kmplayer and Kaffeine folders in
/home > .kde4 > share > apps
The I restarted Kmplayer, which now plays as normal for all types of video files
( *.flv, *.mpg, *.mp4) that I tried it on, and audio files (*.ogg, *.mp3).
However, Kaffeine is only partially working.
Kaffeine will play audio for sound files (*.ogg, *.mp3) and will now show the video
for video files ( *.flv, *.mpg, *.mp4) but won't play the audio from the video files.
Ah, so close. At least Kmplayer is working again.
Even though I use VLC almost exclusively for media files, it still irritates me when
something like this happens and things don't work as they used to (and are supposed to).
Well, gremlins still live in them there bits.