Unfortunately with drive failure you are not guarantied to get the data back, and there are a number of different causes for the failure, but you can some steps and hopefully one of those will work.
Try connecting the drive to another computer if you have one available, if not try a different port as the issue might be with the connection not with the drive.
Remove the drive from the enclosure and try connecting it to your machine. You can get a drive adapter or if you have a desktop connect it as a secondary drive.
If the external drive was setup as bootable , try connecting it to a Mac, is it will bypass the boot partition.
If you see the drive when you connect it to a machine, but can't get the files. Try some software recovery tools like getmydataback.
If all else fails or you initially thing that the problem is with the drives hardware, you can ship it out to a recovery service. There are a number of them, but they can get pretty pricy (could be more then a couple grand). Before shipping it out make sure to find out the price, and see if you are willing to pay the amount for that data.be aware that they'll do their best, but they can't guarantee that they can recover the data