I appreciate the detail in your post. One thing I'm not clear on is the last bit where you were having trouble changing it to be writable with the "remount command failed" thing. This could possibly be due to some error encountered as a result of changes you or Knoppix has made in the current session rather than a persistent problem. Did you try completely rebooting, then immediately mounting sdc1 (without any intermediary steps like you described in your post), then changing the read/write mode? I have had very similar errors crop up, and this often turned out to be the problem. Perhaps trying to change the read/write mode on an unmounted drive made some process hang without relinquishing control of it. You seem pretty knowledgeable from your post, so I don't mean to patronize you, but it's a simple step you might've missed.

If that didn't work, I've had a great deal of problems trying to mount/umount or otherwise manage devices just based on how I try to do it. For example, using the desktop icons that are automatically created for each of your devices sometimes fails to register changes I make to it, or they return errors very much like yours. Sometimes adding and using the "Storage Media" applet to the KDE Panel (the equivalent of the windows start menu/taskbar) will circumvent this problem or may have that trouble itself. Sometimes opening a console and using the text command "(u)mount /dev/sdc1" works for me, other times i must use "(u)mount /media/sdc1". I have the most success using one of these console commands rather than relying on the GUI. Like I said, this doesn't sound like a persistent problem that must be solved before you can move forward with your data recovery; it sounds like a fairly common error that I often encounter that is technically "your fault". If you've already tried the methods I've listed here or they end up not working, try opening a console and typing ps -aux. From this list try to identify anything that might be hanging or that should not be running/should have been closed. Use the "kill" command on the corresponding PID(s) and then try again.

Just as a rather recent example, I tried to unmount my second flash drive partition after I finished watching a video, but I ran into the same problem you had. It turned out that when I closed Kaffeine (a media player) normally, it didn't actually fully shut down. When I noticed that and killed the process, it allowed me to unmount/otherwise change that partition again.