If you have not done so, try a reset (hold down the Home and Power buttons until the Apple logo appears).
Try un-pairing and repairing your keyboard in Bluetooth settings. And, of course, make sure you're batteries are good in the keyboard.
Not sure about Game Center, but the iOS 9.01 update is out to day. I don't think it said anything about Game Center fixes, but it might be worth a try.
That's all I can think of right off the bat.