I had the same issues as everyone with iPad 2 which my sister gave me with iOS 7 Factory resetted and I did not have the ability to Delete the Facebook Account (which was never set up on here in first place) OR change the User ID whatsoever. After trying this, it did not work.... Settings > General > Reset > Location and Privacy.
However... I did find finally a solution for myself and maybe you guys can use it as well.
*This is really my first hands on Apple iOS experience as I'm a hardcore Android fan and while there are things that are nice on Apple, I can't stand not having a Back Button and silly issues like this which if you see on Android is not the case with Jelly Bean or newer, especially Galaxy S4.
1. With an Ipad 2 with iOS 7.1 Upgraded. Backup to iCloud and also manually backup to newest iTunes (I used iTunes 64bit on Windows 7).
2. After properly ejecting, Settings > General > Reset > Reset Location and Privacy.
3. Reset Home Screen Layout
4. Reset Keyboard Dictionary
5. Reset Network Settings. At this point it should automatically restart.
6. After restarting, Go to Settings > General > Reset > and "Erase All Content and Settings". Make sure Not to tap the first option "Reset All Settings".
7. It will restart and act like a new iPad but here's the KEY that I think made it work. When it asks you whether or not to restore from iTunes or iCloud, you say set it up like a New iPad. You still can restore things from the cloud afterwards. I think if restoring from a backup/iCloud, you will have the same issues. When I did this, and afterwards put my iCloud information I found immediately under the Settings > Facebook > that I was able to fully type in my Facebook email and password and it Worked!