The symptoms indicate either a failure in the enclosure's USB interface, or a failure in the drive itself. Since the system does not recognize that there is a device attached, the probability is that the USB interface has failed. This seems to be the usual failure mode of external hard drives, in my own experience.
There are two major options available to you at this point. How you decide to proceed depends on how comfortable you are with computer hardware.
The first option is simplest. Take the unit to a local computer shop, explain what happened, and ask them to extract the hard drive and check it in a new USB drive enclosure. They will crack open the case, extract the drive, and put it into a new case. They will then test the drive to see if it can be seen by a computer. If so, you have the same drive back in a new case. If not, they can proceed to diagnose the drive for problems and advise you on how to proceed from there.
The second option is cheapest. Purchase a universal IDE/SATA to USB kit on ebay. For about $10, you get a drive-to-USB converter cable, a power supply, and SATA cables to connect to a SATA drive. Carefully open the case, extract the drive, and attach the converter cable and power supply. Plug the cable into your computer, and if the drive is still functional, it will appear as an external hard drive. You can then decide whether to install it in a new case, or purchase a new external drive and copy the data from the old one to the new one. If the drive is not functional, then it is time to call the data recovery professionals.
Regarding taking the unit directly to a data recovery house ... They will gladly extract the drive from the case and check it out, but if the drive has not failed this will be an expensive way to find that out. Their services are necessary if the drive itself has failed, but the other options are less expensive and should be tried first.