You say you "ran the software" but it could not find the troubled drive. I suspect you used the WINDOWS version of the software. That may not be able to test a device that Windows does not recognize.
Go back to the Seagate website and download the Seatools for DOS package. Get the version for a CD-ROM - it comes in two choices. One is the .iso image file. The other is just a compressed .zip file that, when unzipped, creates the same .iso file. Once you have the .iso, then you need a blank CD-R, a CD burner, and disk burning software that can make a bootable CD from an .iso image file - like Nero, for example. You use that to make your own bootable CD-ROM disk that contains Seatools for DOS.
Place the prepared disk in your optical drive and make sure your BIOS is set to boot first from the optical drive before it goes to the HDD. It will boot from this new CD-ROM disk and load a mini-DOS into RAM, then start up a menu system from which you run all its diagnostic tests. The huge advantage is that it works even with no functioning HDD in your machine, AND it works without using Windows at all! So it CAN access devices like HDD's that Windows cannot communicate with, and run tests on them.
There are several non-destructive tests you can run to determine whether there are any hardware problems. There are also somew tools that try to fix things, and certain of those DO destroy data! Anything like that will put a big warning message on the screen and ask for your permission to proceed. So WATCH THE MESSAGES and do NOT let it do anything to over-write or destroy existing data.
If this tool tells you there are no serious hardware problems, then it's software - as in, corrupt data making disk access impossible - which is more likely to be the case from your story. Re-run Disk Management and look in the LOWER RIGHT pane. It SCROLLS so you can see all its contents. Your troubled disk ought to be there among the hardware devices, even though it may NOT appear in the Upper Right pane. Look at its information, especially the File System it says it has. Normally this is NTFS. But if your says "RAW", that means some info in the Partition Table or the NTFS Sile System tracing files is corrupt and Windows cannot understand it. In that case, look around the web for ways to recover from a RAW Format HDD.