Well... I certainly understand your being upset. Tons of files and work all just beyond your reach.

Lets try this:

Disconnect everything from the iMac (except keyboard & mouse).

Connect LaCie drive via USB cable.

Turn on drive. Turn on iMac when drive is up to speed.

When System Prefs "sees" the drive, fire up Disk Utilities.

Can Disk Utilities "see" the drive?

 Don't sweat the petty things, and don't pet the sweaty things!  ; )