What kind of drive is it?
Is it a USB drive?
Back -- WAY back -- in the old days of SCSI, Apple's HDSC setup utility offered the user the ability to "write a new set of disk drivers" to a drive that was having problems.
However, there seems to be no way to perform a similar operation with USB drives (at least not one that I'm aware of).
If the external drive's disk drivers get corrupted through an "improper ejection" of the drive, and they won't mount the drive again afterwards, there seems no easy way to repair this.
It's also possible that the drive's directory got corrupted during the force-eject.
Again, this might prevent the drive from being mounted on the desktop.
There are some apps out there that can do some disk repairs, such as:
TechTool Pro
and
Drive Genius
and a few others.
One of them might work, but then again, you might try all of them and still be "where you are right now", with an un-mountable drive.
In that case, you may need to go to a data recovery app (these are different from the repair apps).
A data recovery app is designed to "work around" the directory, and go "right to the disk platters and sectors". Then, it scavenges around to find whatever data it can, attempts to re-assemble the data, and saves what it finds onto a "scratch" disk. You'll probably lose all folder hierarchies, and many (or all) file names, but you can often get the data files back this way.
One of the best data recovery apps is ProSoft's "DataRescue".
There is also Stellar Phoenix Data Recovery and Disk Drill.
These applications all work the same way:
1. You download the app in demo mode for free
2. You run it in demo mode to scan your problem drive
3. The app scans the drive, and presents you with a list of files that it discovers
4. HOWEVER -- the app will only let you recover ONE file in demo mode. The purpose is to demonstrate to you that the app will work for you BEFORE you pay the registration fee.
5. You now pay the fee, get a code and enter it, and the app goes to work on the drive and recovers whatever it can.
You'll also need ANOTHER drive which will serve as the "scratch disk" to receive the recovered data.
Worth trying, and get back to us.