I have a complete file system backup.
Is this a back-up - in that you copied over the fiies and folders you wanted to protect - or is it clone of the drive ? If it's a clone you should just be able to reinstate it on the new computer. If it is a file copy back-up then it will not contain the keys you are looking for as they are stored in the registry to the best of my knowledge.
Can you not just re-install Office 2013 from whatever media you installed it before, it will have the authentication key on or with it ? Unlike Windows you can install MS Office on as many computers as you have a licence for. If it is a single use licence you can move it from one computer to another so long as it is uninstalled from the previous one.