Hello disturbed13 and welcome to the forums
This should be relatively simple
The Office 2010 product key is stored in the Software registry hive, however it is in an encoded format and we'll need a tool to decipher it.
I would like you to boot into Linux and open the file manager. Navigate to the following folder on your Windows hard drive:
You should see a few files in here, the ones without extensions are your registry hives. Copy and paste the file "software" (there is no file extension) to a a USB flash drive so we can access it from another computer.
For the next part, you'll need access to a Windows computer.
Now I would like you to download ProduKey (by Nirsoft): ProduKey - Recover lost product key (CD-Key) of Windows/MS-Office/SQL Server
Direct download link for 32 bit computers: http://www.nirsoft.net/utils/produkey.zip
Direct download link for 64 bit computers: http://www.nirsoft.net/utils/produkey-x64.zip
- Insert your flash drive into the Windows PC and run ProduKey.
- When ProduKey launches, press File > Select Source and you will be presented with a new menu
- On this, select Load the product keys from an external Software Registry hive and you will be prompted to open a file
- Navigate to the software hive on the external drive and open the hive
All of your product keys should now be displayed
If you right click on them, you should be able to copy them directly.
I would recommend that you save this list elsewhere just in case this happens again. Encrypt the list and upload it to Dropbox or something