Would a lightning to HDMI cable facilitate this? I've heard Apple TV being the answer, can anyone elaborate?
With iOS9, you can turn on AirPlay and Screen Mirroring to play pretty much anything on your Apple TV (3/4). There's a slight lag though, it's not the same as using your iDevice as a controller to play a game running on your Apple TV 4. HDMI cable would have no noticeable lag. There are a handful of games that will Airplay the main screen, and have a different display on your iDevice, which is pretty neat. Pilot's Path is probably one of the best examples of this. EDIT: Massive thread with details...
You can mirror any game on a TV via cable, but only some games actually format for display on a TV. But being tethered to a TV isn't that great of an experience really.