|Would be useful to know which model you have as there are quite a few...?|
Meanwhile - try a different (known to be be good) cable.
It would appear that some of these drives use the mini usb connector(s) - at least at at the input to the drive? These are not the strongest design... I have read accounts where the actual connection by the socket onto the internal motherboard is prone to failure; depending on quality of the actual assembly - not the least the soldering...
It may be that yours either has a problem with the cable itself, or the socket on the drive housing; possibly even internal electronics.
If testing with another good cable doesn't resolve it, can you you extract the drive from the case? If so then get an usb adapter to enable you connect the drive directly (via usb) to your computer. SATA -t usb cables are quite cheap and ideal for such situations.
I bought this one from Amazon.co.uk for about £8-10: and it's also on Amazon.com for about the same (in $$$).
StarTech.com USB 3.0 to 2.5" SATA III Hard Drive Adapter Cable w/ UASP - SATA to USB 3.0 Converter for SSD/HDD - Hard Drive Adapter Cable
Very satisfied with it.