How to restore from .sparsebundle stored on Time Capsule?

How can I restore the files from one Mac encrypted with Filevault (forgot encryption password but have recovery key) to a different Mac from an encrypted Time Capsule backup?

Note: I no longer have possession of the original Mac (rMBP) so my only means of retrieving its information is to somehow restore its files from the .sparsebundle file stored on the TC to my new Mac which has different hardware.

Original Mac model:

Apple MacBook Pro "Core i7" 2.6 15" Retina 2012 (my last backup was with OS X recently updated to Yosemite 10.10.1 but backups from 10.9.3 should also still be on TC)

Current Mac model trying to restore to:

Mid-2011 21.5-inch iMac (running OS X 10.9.5) I also have an identical Mid-2011 21.5-inch iMac (running Yosemite 10.10.1). Neither work for the restore.

When I try to use Migration Assistant I can see my original Mac as a restore option, but it asks me for the Sparsebundle password. I typed everything I could think of that it might have been when I originally encrypted the rMBP's drive using Filevault 2, but no luck. When I type in the 24-digit recovery key this does not work either. I have kept this key as an emergency means of restoring my drive and I don't know why it will not accept it. I am not afraid of using Terminal if needed if somebody could provide specific instructions to solve this problem. I would be deeply grateful...