Use Airplay screen mirroring to play on your television using an Apple TV (2nd or 3rd gen).

Some changes to GamePlayViewController.m and added some CGSize functions so it is possible to use Airplay for playing on your television. 
In my test cases, the lag tapping buttons depended on the overall load of my home wi-fi, and was unnoticeable when there was zero other network activity. I tested this playing Mega Man 4.