If you wrote down your Recovery Key then you can do it without your other computer. In this article, see the section "Connect Another Computer":

https://support.mozilla.org/en-US/kb/how-do-i-set-up-firefox-sync#w_connect-another-computer

Does that work?

Do you know your Sync account name (the email address you used when you initially set up Sync), the account password, and your Sync Key / Recovery Key?
You need all three to "add a device to Sync".

If not, do you have Sync set up on another device, so you can recover those 3 items?
That information is stored in the Password Manager as two separate "chrome://" addresses on laptop / desktop installations. With mobile devices you need an add-on to see the Sync Key.

If no for either, you won't be able to restore your data. And without the Sync Key your data can't be decrypted, that Key is an encryption / decryption algorithm specific to your Sync account.

Hey guys thanks for the info. I am not sure if either would have helped because as much as I save stuff, that information was not on my list. The whole thing is mute at the moment because after 10 hours of messing around I accidentally fixed my old laptop. The info you just told me, will be the first thing I save in my list of important info.

Thanks so much

Regular backups of your entire Profiles folder also would be useful, since much of what you see in Firefox is not yet included in the data stored on the Sync server. See Back up and restore information in Firefox profiles.