You have probably had all of the usefull answers to this already - however a few other bits of info that might help.
If the old computer had DOS 6 then it has Interlink - this is a little server program you run on the DOS computer and a client to run on the new computer to remotely "mount" it's drives.
You can get laplink style parallel link cables from Maplin if you need. These work with laplink, Fastlynx, and Interlink. Order code is BJ27E costs a tenner. Failing that you could make your own see here:
Another option not discussed is just using a straight null modem serial lead with a standard BBS style comms package running on both ends and transferring files with ZModem. Hyperterminal will do it on the new computer, and a old DOS package like Procomm, Telix (free), or any of the many varieties that used to come bundled with modems etc.
Back on the topic of moving drives - if you can't get the 5.25" drive in the new case - could you stick a 3.5" floppy drive in the old case as well as its 5.25" drive so you can copy the files that way?