If you don't have your recovery key, then unfortunately you can't sync. Firefox sync is not a replacement for a backup system, and shouldn't be used as such. Unfortunately at this point you only can set up a new sync account to use in the future.

For future backups, set up a backup plan and also make sure that if you re-install windows you back up your data before wiping.

There is nothing in the original post about using Firefox sync as a back up option. I think what is meant is this: The instructions one gets to find one's recovery key are incorrect. For instance, there is not a 'Manage Account' button to click when one does as instructed. i.e. click the Firefox button (top right of browser window in Windows 7); then 'Options', then in the Sync panel click 'manage account'. <<< this is nonsense because all one gets when following these instructions is the links for pairing a device and setting up sync (see attached). NOWHERE can I find a 'Manage Account' button. What is required is up to date and correct instructions on how one can get a copy of one's recovery key - which, as it happens, I have never found yet. Information is not being kept up to date so users get frustrated. I want to add another PC to my sync account and cannot because I am continually misinformed on how to do so. It is complicated enough to begin with because no one has bothered to EXPLAIN the steps clearly. This needs to be sorted before I have to start opening email accounts JUST so I can sync my Firefox information between my computers.