The plugable.com dock should do the trick, IF the old drive can be mounted on the desktop.
Even if it can't be mounted, there are still tricks you can use to get data back.
Will the Apple tech give you the old drive back?
That's the first step.
Once you have the drive "in hand", buy the dock, and try mounting the drive that way.
If the drive mounts, and you can "see" the files and folders you want, try copying them to your new internal drive (or to any other drive).
If the drive DOES NOT mount, you could try this:
You might try running DiskWarrior on it. Bear in mind that DW is an app that is designed to work on drive directories ONLY, and it can do a superb job repairing and rebuilding them. DW might be able to "get the drive mount-able" again.
But also be aware that DW cannot do "data recovery", nor can it repair physical drive damage.
If DiskWarrior doesn't work, here's a last-ditch procedure that might:
(Warning - no promises of success)
1. Re-initialize the old drive (yes, I said re-initialize)
2. Do not, DO NOT choose the option to "zero out data" (you DO NOT want to do this, Kabish?). The object is to replace the old directory, but leave the data on the platters "untouched".
3. If the drive now mounts (but shows as "empty"), use data recovery software (DataResue3 is one of the best) to scavenge the drive and find the old file data "out on the platters).
4. DR3 will locate and reconstruct old files, and then can save them to a "scratch drive" (must be a drive other than the problem drive).
5. It's important to understand that in the data recovery process, you may lose file names and folder hierarchies. This is to be expected -- the important thing is that you get the _data itself_ back.
I had a drive with a corrupted partition that had hundreds and hundreds of mp3 files I wanted back. Nothing seemed to be able to repair the partition or make it mountable. So I reinitialized the drive (as above), then used DataRescue, and got the files back!