Can't login to iCloud on Apple TV 3rd gen

I just had to restore my old Apple TV (A1427), and now it will not let me login to iCloud. It asks for my username, then password, and then I get the security code on all my other devices. I enter it on the Apple TV, and then I get the error that my password is incorrect, and that I need to reset it. My password is not incorrect, since I was able to login to the iTunes store just fine, as well my iCloud account on my computer just now. And also, when I entered the password, I got the verification code on all my other devices.

I know that in older versions of the tv software you had to go back and append the verification code to the password and enter it again, but this time I was always being presented with a new screen specifically for the verification code after I entered the password. I've tried this about a dozen times now, with the same result - any advice? Thanks!