Fully signing out of Game Center and then signing back in can sometimes fix issues issues such as not being able to connect your game to Game Center. Please try the following steps:
1. Sign out of Game Center from your Settings app.
2. Fully close the game as instructed here:
3. Restart your device.
4. Sign back into Game Center from your Settings app.
5. Sign into Game Center from within the game.
6. If the previous steps don't work, repeat steps 1-5 while using a different internet connection.