I have followed all relevant steps, my iPhone5 and apple TV are connected to the same wireless network, but I still cannot use airplay. When I choose the airplay option on my phone, the only option I get is "iPhone"

Welcome to the Apple Community Willy.

Try the following steps, check whether things are working after each step where appropriate, before trying the next.

1. Check AirPlay is turned on on the Apple TV (turn it off and on if it already is)
2. Check that both devices are on the same network (Settings > Wifi, on the mobile device and Settings > General > Network, on the Apple TV).
3. Restart the Apple TV (Settings > General > Restart).
4. Restart the Apple TV by removing ALL the cables for 30 seconds.
5. Restart your router. (Also try removing it’s power cord for at least 30 seconds)
6. Restart your mobile device.