Once upon a time not too long ago, my brother's external HDD died. It simply wouldn't be read by the PC with the same error (device not recognized). It was in warranty so we could send it to the company (in Ireland, it seemed). It however was showing the HDD light sometimes and the disc was definitely moving (you can sense the rotation of the platter). So I opened up the HDD instead (voiding the warranty of course) and connected the 2.5" HDD inside into my desktop. Voila! working again..
This might be a one off case but for me, when the HDD was working but not being shown by the PC (even after checking on different ports and different machines), it was a problem with the Sata to USB interface that the external HDD was packed with.. Not sure why it decided to die randomly but it did.
So my suggestion would be to try to get your warranty.. if they can't help you, you can always open it up (seeing as without the warranty, the HDD is worthless anyway if it isn't working so you might as well open it and connect it into your desktop internally).
That's what I would do.. oh wait, that's what I
did 
Might just be me, oh well
Just trying to give you an option in case the warranty doesn't come through, that's all. Hope it doesn't have to come to that :P
Good luck!
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