A bit of advice before going further:
In the future, if you have files that are important to you, you would do well to keep them in MORE THAN ONE place. To avoid problems like the one you're experiencing right now.
If you can access the music files via a media player, what that suggests is that the drive is still "physically good" (no hardware failure), but you have some kind of directory damage.
It's -possible- that a directory repair app (such as DiskWarrior) could rebuild the directory and make the drive accessible again. There is also "DriveGenius" and "TechTool" that may be able to do this, but for disk directory repair, DW is "the champion".
But sometimes even directory repair software won't work.
That said, I was once in a similar position.
I had a bunch of mp3 files on a partition of a hard drive, and suddenly that partition could no longer be mounted on the desktop.
Nothing "simple" worked.
The partition in question just wouldn't mount.
The files weren't invaluable to me, I just wanted to get them back.
Here's what I did:
First, I got ahold of "data recovery" software (DataRescue). Note that this is not "directory repair" software -- we're now into data RECOVERY.
Next, I re-initialized the entire drive (yes, you're reading that right -- RE-INITIALIZED the drive). I just used one partition.
After doing this, the drive now "mounted" on the desktop again, but it showed up as being "empty" (i.e., no files on it)
Then, I aimed "DataRescue" at it, and let it do its thing. If I recall, I set DR up to look for "mp3" files.
DR "scavenged" the drive, found the files and saved them to another (scratch) drive.
WHY it worked:
Normally, when you re-initialize a drive, the drive's directory is "wiped clean" and replaced with a fresh "empty" one.
BUT -- the actual DATA out on the sectors of the drive itself is left UNtouched.
Data Recovery software (like DataRescue) can "bypass" the directory (which of course shows "nothing there") and "go right to the platters", scavenging what it finds and then re-assembling the data onto a scratch drive.
You WILL need ANOTHER DRIVE to serve as your "scratch drive".
You're probably going to lose many file names and folder hierarchies. This is "par for the course" when you're doing data recovery.
A trick that worked for me with mp3 files that no longer had "finder names":
I took a spare drive, put an OS onto it (clean and fresh), and then put all the recovered mp3 files onto it.
Next, I opened iTunes (which was new and as yet had no music library).
I then imported all the recovered files into it.
Because the metadata of a music file is actually "part of the file" (not the file -name-), iTunes was able to read the metadata and re-group most of the files into artists, albums, etc.
I still had to rename files individually in the finder, a lot of work.
But again -- I GOT THE FILES BACK.
This takes work and time, and some money.
But do you want the files back?