External Hard Drive won't mount. How can I transfer the files?
There are two possibilities. 


The disk itself has failed. You now need a high-powered recovery program like Data Rescue, or a service like DriveSavers. Cost is about $100 for the first and $1000 for the second.
The case has a problem. Given that it's a Buffalo case I think this is more likely (their products are rather low quality). Get a new case or a drive dock, open the old case, remove the drive and put it in the new case / dock.