Sounds like the 5.25 inch drive died. If none of the 10 floppies will format or read.
They can be brought back to life, if the mechanics are all ok and it's just dirt on the read heads.
It is not easy to damage a 3.5 inch drive plugging in the wrong way, but I am not so familiar with breaking or fixing 5.25 inch. It is probably easier to take the heads off for cleaning, if you can't get another one.
But, why not just boot from a floppy on the 486, from 3.5 inch PC drive, and just try it that way? Crash bang wallop, read the data off onto another hard disk, on a different controller. Job done.
A floppy disk is a floppy disk, after all. From the computer's point of view. It does not care where it boots from.
I would not boot with the whole diags tool, that sounds a bit risky unless you know what you are doing and are confident you will not damage the data. Practice a bit with it first, with the hard drive disconnected.
If you get confident with doing this sort of stuff, with IDE and SCSI too, you can earn some very nice money. Data is VALUABLE. Only now are you realizing how value it can be. Top dollar at this game is 10% (finders fee) on the value of recovered data. You got to be VERY good to charge that though. No recovery of at least 80% of data, no fee, is usual. Clients want the whole thing or they are not happy.
I usually just charge for materials, but rarely offer the service. It can be a little dangerous. You don't know WHAT is in the archive to begin with, if it isn't yours. Could be anything. Usually it's totally harmless.