Wireless connection: iPad <--> Projector

Most of the classrooms I teach in have ceiling-mounted projectors. There is a looooong HDMI cable that snakes along the ceiling to the wall and down to a desk where there's an Apple TV. This is great because I can walk into the room, mirror my screen and start presenting effortlessly.

Here's the problem. Most of the other teachers use a laptop, so they pull the HDMI cable from the Apple TV and plug it into their laptop so that they can present. Over time (5-ish years) all of this plugging and unplugging half a dozen times per day has taken its toll on the Apple TV's HDMI port, the cable, or maybe both. It can take 10-15 minutes fiddling with the cable to find that magic sweet spot where it makes a good connection with the HDMI port in the Apple TV. The Apple TV probably wasn't made with all that action in mind.

My question is this: is there some device or dongle that I can plug into the back of the projector and just LEAVE there? Can the iPad screen cast to anything other than an Apple TV?