Unable to sign in to my iCloud account after changing password

I changed my Apple password because I keep getting the iCloud login screen on my PC which doesn't accept the old password.

Now I've changed the password, the iCloud sign-in screen keeps coming up asking for me to login, sends me the verification code, which it accepts, then asks me for my password again.

It seems to be caught in a loop.

[Re-Titled by Moderator]

Original Title: Changed my password. Now I can't sign in to iCloud