I've finally found the time and the energy to try out part of Xepher's suggestions. Today, I bought a IDE-to-USB drive housing thingy, figured out how to remove one of the hard drives from the broken machine, figured out which part of the seemingly simple instructions for assembling the IDE-to-USB housing thingy were wrong and which were impossible to carry out due to shoddy workmanship on the casing, brought the whole shebang to the studio, loaded Kubuntu there and got the drive to work. Yay me! The bad news is that I ended up having some of the same problems that I had with both drives at home: some of the files I knew were there when the computer broke weren't to be found on the drive, and some of the others seemed to exist, but when I copied them, Konqueror refused to do so. With these, all I get from copying the files is 0-byte files on the target drive, even though Konqueror lists these files as available and having a believable file size on the source drive. Weeeeird. Opening these files in, say, OpenOffice (I am mainly concerned at this time with my financial accounts as doing my bookkeeping is enough of a pain without having to reconstruct my old invoices. Anything else including music files and even my backed up art images is of secondary importance - that stuff exists on CDs or can be remade easily), I get an error saying the file is corrupt, can't be repaired, sorry, can't open it at all. Strangely, some PDF files I made of several of my outgoing invoices did still work, and these have now been copied on my portable drive, leaving me with only three invoices to reconstruct or otherwise recover. So it's not all bad.
I'm going to do the same with the other hard drive. That one's harder to reach inside the PC's case, but it may have more functional data left on it, including those missing bills and an old copy of my email folder.