When I changed to an SSD in my MacBook I backed up all my applications to a time machine back up and then restored my applications and settings from this backup onto the new SSD. Probably doesn't help you unless you can back up the old HD and do it that way?
I backed up my entire system to an external HDD using time machine but it makes no difference with MS Office 2011 apparently. It just knows you have done an upgrade of some kind and demands the product ID.
You have to be able to access the hard drive you migrated from.
Make sure all Office programs are closed.
On the new hard drive/computer you migrated to, delete the following file:
Go to the HD you had a working copy on, and copy that same file from the same location to the new HD.
It should have an older date from when you installed it and activated it.
Did that having read it online and it failed miserably. From what I read, I am basically stuffed. My last chance is to ring MS and quote the ID's from Word etc. Failing that, a new MS Office for Mac is due out in the autumn apparently.