"My KDE swap files are on the external card, and I've left them there."

That is the problem... PB's way was to make a small partition on the internal card for swap. Your system can't run without the external card now!
Not sure how to remove a swap with it running, but power down, remove the external card, and power up. You will get some complaints about running slow, etc. But it will run. The KDE is OK, I bet.

Mike.