This app was mentioned in 5 comments, with an average of 1.40 upvotes
Well, there are several sources saying that it works with an AirPlay receiver app on Shield. For example https://play.google.com/store/apps/details?id=com.splashtop.m360 My question was only about the quality and limits of such connection.
To cast my photos and videos from my iPad to my Nexus Player I use "Photo Video Cast" from the App Store.
Also, if I want to mirror your iOS screen to the Nexus Player, You can purchase Mirroring360 from the play store. It let's any Android device act like an AirPlay receiver. https://play.google.com/store/apps/details?id=com.splashtop.m360&hl=en
There's an AirPlay receiver now for Android TV, which while not Chromecast is still something.
https://play.google.com/store/apps/details?id=com.splashtop.m360
Disclaimer: I couldn't get it to work on the Nexus Player, was great casting an iPad to my HTC One though. I got it working once on the NP by sideloading the free version of the app back in December, but haven't tried since then.
I use Mirroring360 receiver to cast my iphone to my Amazon Firetv, it is installed on my fireTV and I just use airplay on the phone. Works great!
https://play.google.com/store/apps/details?id=com.splashtop.m360
There is this app, https://play.google.com/store/apps/details?id=com.splashtop.m360