Doesn’t let me sign in to iCloud

When using 2 factor authentication on an older device you need to enter the password and code together, and there is a time limit. To make the best use of that time enter the password, then request the code, and add it to the end of the password

Get a verification code and sign in with two-factor authentication - Apple Support

Have you tried turning the Apple TV off and after a couple of minutes, back on? Is there an Internet connection on the TV?