I'm not sure. It really sounds like there is a problem with the partition table or a file table on the hard drive. Those USB connectors usually work well as long as all the connections are good. I use one regularly for data recovery on client's hard drives.
You could connect that hard drive to any desktop computer as a secondary drive and Windows should read it, as long as the drive is good. Try the connector and external drive on someone else's computer and see if you have the same problem. Or test a different external hard drive with the cable if you have access to another drive. That will help isolate the problem to the cable, drive or PC. My guess is your drive.
If no dice, move on to the free partition or data recovery software since you have nothing to lose. Maybe someone else will chime in if they have a better idea.