I have found that, with external drives in a largely stationary environment, it is far more often a failure of the enclosure than the drive itself.
New enclosures for hard drives are readily available and don't cost very much. But that is not the first step I would take.
What I have done is to obtain a SATA/USB converter from your local electronics store -- these are pretty cheap, and the one I have found that is both most reliable and easy to use carries the "Apricorn" brand. You are going to need to break the drive out of the enclosure in any event, and some of these enclosures are pretty tough. They are seldom assembled by screws or any other method that presumes that you'll ever want to get inside, so you will need something to penetrate whatever small seam you can find and pry it open by force. This will likely break the case, but so what? It's broken already.
After you've gotten the drive out, use the adapter, which will plug into the wall, into the two SATA connectors on the back of the drive (one for power and one for data), and plug into the USB port on your computer. After you get it plugged into your computer, you should see it as a removable drive, and you should be able to obtain the directory listing.
Assuming you see the drive (if you don't, it IS the drive and you're screwed), now you can do whatever you want with it -- transfer the contents to your internal drive or a new external drive, or install the drive as a new internal drive on your PC (assuming you have a SATA cable, an available head on your motherboard, and an available SATA power plug on the cable coming out of your power supply.