I would suggest to connect the drive onto a separate IDE cable, say remove the CD Rom drive and put it on there. (you can run without the CD drive for a while). This should reduce any data errors affecting your system drive and OS operation.
Then use a tool called "R-STUDIO data recovery" from
http://www.r-tt.com
It is a fantastic tool which allowed me to recover my whole email server store when my server crashed. It is not a free tool, but well worth the cost and once you have it, you always have it.
Changing PC's should not affect whether the HDD is visible in the other machine if you have a fairly recent BIOS.
The problem with the drive might be that the File Allocation Table (or FAT) became corrupted (If it is a FAT file system formatted drive). The drive's partition information also may have been affected. This may have been caused by a "Badspot" on the physical media of the disk.
This utility reads drives and file systems that are even not recognised by BIOS ! (and no, I don't work for them as a salesman :)