It would very much depend on what is the problem with the drive, if it is mains powered, have you checked that it is not just the mains adapter? If it seems like it is powering up, have you checked that it is not just the USB cable? Alternatively, if you are comfortable with a screwdriver, you could try to get your hands on a second hand working drive, and take out the disk from the broken one to the other one.
lastly, you can leave drives in to have data removed, but it is damned pricey.
I think first off, you need to let us know exactly what model of drive you are talking about, and what the problem with it seems to be.
@Jodav, does it not sound like it is a portable drive as opposed to an internal HDD???