I have put together many DOS base computers back in the day. My first DOS system was a 386-40 with an IDE hard drive. Reused the hard drive (50meg quantum) from my Amiga 500 system. Still got that old drive in my closet.
I came across an old 286 system that had a working MFM hard drive in it. (think it was 20 megs) I forget what version of DOS that was in it but it was an older version. Disk compression was popular at that time so I installed that with DOS 6 and got a nice increase in size. The point is that I didn't need any drivers to install to run it.
So you have a A2000 with an 8088 bridgeboard in it with a MFM card installed in one of the ISA slots running a hard drive. Your trying to get your information from the drive.
The A2000 is making it complicated. Take the controller card and hard drive out and install it in a PC compatible. The BIOS on the controller card will help the system detect the drive.
You need to make sure it is working. If you don't hear the hard drive it is stuck. (You will hear it if it is spinning) The heads are stuck to the platter. That happens when it sits for too long. Assuming it works; you can use a boot disk (DOS 6 will work)to boot the computer and read your info.
If you get it running I would just copy the complete drive to something more modern.