* * * * * Problem solved * * * * * *
I came across another post where someone was having the same problem as I was, but they were using a USB Blaster cable. Their cable wouldn't appear in the hardware listing and they kept getting the same TJAG server error that I was. Their situation was that they had been using Quartus v10.1 SP1 with no problem, but only had problems when they upgraded to v11.0. When they reverted back to v10.1, everything worked again. Somebody suggested that Altera missed out some files in some of the releases.
Hoping my problem was also caused by missing files in v13.1, I uninstalled it and installed v10.1 SP1 instead, and everything worked first time, even with my old ByteBlasterII cable.Maybe there's a newer version that would still work for me, but I'll stick with what works.
So, if anyone else has problems with cables not showing up in the Hardware list, or get JTAG server errors, try installing an earlier version than the one you're using.