A failed hard drive is very difficult to diagnose, especially if it can't even be seen by one or more computers. Since it's an external, and it's in an enclosure, it is possible that the enclosure is faulty, and not the hard drive itself. If this is the case, then you would need to borrow, or purchase another enclosure just to test it. If you know anyone who has a similar enclosure, perhaps they can assist you. The USB cable, or whatever type of interface cable it has, could also be faulty. Have you tried using a different cable?
Do you happen to know what model number/manufacture makes the drive? I'll assist you in finding what kind of drive may be in the enclosure, whether PATA or SATA.
As far as data recovery if the drive has actually failed, there are professional companies that do this kind of work. They are expensive, probably in the $100 -$400 range.