Access encrypted harddrive from broken computer

My main computer for work has broken down! The computer itself can of course be replaced but a lot of the data can't (*)

I'm using my Trisquel machine to try to reach the data, but I run into the problem of accessing encrypted data. I've searched for answers but can't seem to find some useful ones. For example I've tried this

https://askubuntu.com/questions/196233/how-to-recover-data-from-old-encrypted-hard-drive

but without success. This is the first time I face this problem, so please bear with me...

(*) I AM doing backups but my computer has been working on this for a long time, and it ain't done. I'm using SpiderOak to make a backup of all my work data, which is >2GB. Unfortunately I had made the mistake of also backing up system files. Bottom line is, that the data to be backup up just increased and never finished. Some of the data is still not backed up.