A simple Recovery may still be possible but it will require lots of time and some Specialized Equipment. Generally speaking it's cheaper to pay a Specialized Data Recovery Service to do this work.
But if you have already messed with the Drive and written anything to it or the Data Recovery Apps that you have run have made changes you may be looking at some form of Forensic Recovery being required and this will increase the costs by at least ten fold.
If the Drive [b]Has Not[/b] been altered you need an Application that rebuilds the Partition Tables as these are most likely corrupted by unplugging the Drive without using the [b]Safely Remove Option[/b] under Windows. Unplugging any USB Drive just removes the power from it but does nothing to park the heads before the drive spins down. There is even the possibility of the Heads making contact with the platters if the drive was moved before it was allowed to stop spinning.
Turning off the computer with a External Drive plugged in is [b]Exactly[/b] the same as unplugging it as the drive [b]Is Not[/b] correctly shut down. You can make the situation even worse by leaving the drive connected all of the time so that when the system is turned on there is power but no control of the drive so the chances of damage are considerably greater. These External Drives should only be used when required and always stopped before unplugging. If you leave them connected all of the time the Heat Generated by the Drive running will shorten it's life considerably and place any Data on the Drive at [b]Extreme Risk.[/b]
If you need a External Drive running all of the time or for long periods you need a Actively Cooled Enclosure like the Antec MX1
http://www.antec.com/Believe_it/product.php?id=NDc=
These have a 80 MM Fan in them the same size as used in most Power Supplies to just keep the HDD cool as well as their own Stabilized Power Supply.
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