Apple wireless keyboard 2009 - Apple TV 4

Make sure the keyboard is unpaired from the previous device you used it with before trying to pair it.

Use Bluetooth accessories with your Apple TV (4th generation) - Apple Support

You can also try a forced re-pair:

With the keyboard off, you hold down the power button on the keyboard and the LED will start to blink, do not let go of the power button. Select the keyboard and pair it, when it says "Connecting" you can let go of the power button

If you have other iOS devices, as an option to hardware keyboard, you can use Apple's Remote app, to both type and control Apple TV