Apple TV doesn't play iTunes

I am trying to play the music I have purchased through iTunes on my television, using Apple TV. When the Apple TV home page appears, I select "Music" / "Purchased". A list appears on the right half of the tv screen with topics starting with Shuffle Songs and including Music Videos, Artists, Albums, etc. When I select one of the categories, I see a list of the music that I have purchase. If I then select anything - an album, a song, or Shuffle, and then select Continue, the following message displays: "iTunes Store Password - Enter the password for [my Apple ID]. Using the Apple TV remote to select from the letters that are displayed on the tv screen I enter my password and then select Submit. Then on the TV screen the message says "No content was found." Almost simultaneously, my MacBook displays a dialog box showing that my Apple ID is being used to sign in to a new device. So I select Allow, and a verification code is displayed on my MacBook. But I don't see any place on the TV or on my MacBook to enter the verification code.

I have played music from my iTunes library in the past without any difficulty. However, I recently changed network providers, and had to reconfigure all my devices to connect to the new network. I have done this with Apple TV - it is connected to the network. I use it to watch Netflix and other channels. Only iTunes seems to be a problem.

Can anyone tell me how to correct this?