Yes, Laplink would be very useful.
For the benefit of OP, Laplink was originally produces for connecting PCs to laptops, and transferring files etc. It was originally DOS software, although later there were WinDoze versions. The package came with two cables, one for using Serial ports, the other for parallel ports, the latter was certainly faster, but if you're not copying Mbs, then the serial one would be fine, and the serial cable is easier to make yourself from info you can find on the web.
The software that needs to be running on both machines linked lets both machines see all the files on both machines, and copy files back and forth, indiv and in bulk, between the machines. There is an option that one machine can load the software onto the other thru the link, but I've never needed to use that. The one catch in your present situation is that you need to have booted both machines, so the first step still is to get the machine with the 5.25" drive booted. AFAIR.
If you're a 'purist', then you may want to use the 'correct' version of DOS on both machines, but I'd suggest you worry about that AFTER you've got the second machine running/accessible - ANY version of DOS will do initially.
Geoff