Try connecting the drive to a different USB port and or different computer.
Failing that, try removing the HDD from the case and connecting it to the computer by SATA cable to see if it is recognized that way. If it is, you should be able to recover the files as well. If not then the drive itself is damaged and nothing short of a data recovery service will get the data back and the drive is not repairable.
If the drive does work when plugged in directly, then you can try to put it back in the original case and connecting it again, that might reset things, however, more likely the USB controller in the case is damaged and you will need to replace the case. Fortunately those are quite inexpensive. Newegg.com (and many others) have decent ones for as little as $12 and up. The better ones offer more features such as USB 3.0, eSATA, and Thunderbird connections, for a price, of course.