|Accord to TrueCrypt there is no crack or backdoor password so any program trying to work with file recovery would not be successful I don't think.|
That said, I would try booting to a live version of Linux. Puppy is small and can be run from a CDR. Linux may be able to read the drive. Get Puppy from the link below.
The partition needs to be initialized. I haven't found a method to initialize without formatting.
A quick format does not actually overwrite data, just changes the MBR on the disk.
So, while risky, I think your only recourse other than professional recovery is to format and then use file recovery software.
Once the drive is initiated Easeus or testdisk may be able to repair the partitions and you would be whole again. This assumes the disk is not defective.
Is the external under warranty?
Post the brand and full model of the external drive.
I think there are cloning programs that can clone that hard drive. You would need an exact duplicate of the hard drive itself, not necessarily the enclosure part. That would provide you with redundancy so anything tried now would not necessarily be the end of the line.
How much time, effort and cash you put into this is dependent on how important the files are.
I think you have now learned the importance of keeping at least two copies of any files you need to have.
You may also wish to run a drive fitness test on the external to see if it is damaged. Get the test from the manufacturers site.