I figured out how to fix this. You must go to your registry and delete anything that is related to Facebook (the website's already not working for you... so you have nothing to lose).
All you have to do is Click on Start, and in the Run field, type Regedit (you should be careful when editing the registry, so be aware); Next, click on "Computer" at the top, then click on Edit, select Find, type the word Facebook, and then delete anything that it finds when you search. Think about it... your computer worked fine before you ever visited Facebook.com, so deleting these registry entries that are Facebook specific will not affect your computer.
After you delete one, click on F3 to find the next one... and keep doing this until the search is complete and there are no more traces of Facebook found in your registry.
Then, close the registry, open your browser (ie: Internet Explorer), clear-out your cache (Tempory Files) under Tools-Internet Options, and then go to the Facebook page to login.
I got signed-in on my first attempt... and it had been months since I've been able to get logged into Facebook on this laptop (however it did worked from other computers).
If you are unsure about deleting things from the registry, ask somebody who knows what they are doing. However, the steps above should help you out if you want to give it a go on your own.