3rd Gen Ipad is stuck in a loop and nothing helps

Good day, everyone! I was gifted an old 3rd gen iPad. It's still in perfect working order, but it keeps asking for my Apple account login code in an endless loop. Type it in (perfectly correctly), and it just asks again. It's stuck on iOS 8.4, but I can't update it because I can't log in. I wanted to disconnect it from my Apple account and reconnect it again, but you can't disconnect it from the apple account without first logging in, and I can't log in. It's making me crazy. It won't let me use it in any way, it just keeps asking for a login.

Edit: It displays my email address in settings as if I'm logged into the account, but it grants none of the 'logged-in' functionality because it isn't actually logged in. It has never accepted my password. So it's stuck in limbo between logged in and not logged in.

Model number is: md367s0-a (seems this is 3rd gen)