Blank/Black Screen on AppleTV when mirrored from iMac
I had a similar issue while using an Apple TV to connect a TV to my MacBook Pro (via LAN) and playing around with the different display modes as it suddenly stopped working.

Unfortunately, I don't remember any details any more but (after several failed attempts) I solved it by explicitly selecting Off in System Preferences/Display/Monitor/Dropdown Box Airport-Monitor (maybe detaching and reattaching the ethernet cable and/or turning off/on Apple TV were also involved) and reselecting Apple TV again.

This might be important, as deliberately selecting might have reset and/or stored parameters, which may explain why some people experience negative results.

Sorry, if this was too less of information.