While trying to mirror a display from an Apple device, the Apple TV may be missing from the list of available devices.
To start, turn on Wi-Fi and Bluetooth on your device and choose an Apple TV from the menu.
Additionally, try disconnecting from your Wifi router using airplane mode or similar method and reconnect to the same wireless network. The subsequent attempt should populate the remaining available Apple TV devices.
It may be necessary to restart your device or the Apple TV if the destination Apple TV remains unavailable.