My Apple TV can see my iTunes library, but can't connect to it

My Apple TV stopped connecting to my iTunes library last week. I do not know why. However, my Apple TV connects to all networks services just fine, including apps like Netflix & Crunchyroll with no troubles streaming movies. I've tested the WiFi network. I've rebooted. I've made sure the software was updated. I've logged out of iCloud on both computers, shut off Home Sharing, rebooted, sacrificed a chicken to the binary gods, rebooted, logged everything back in, and still no luck. Everything is as it should be, except the Apple TV won't connect to my iTunes library. This happened on Nov 26-27. It worked just fine before then.

My Mac mini (running Yosemite) can mirror using Airplay Display onto the Apple TV. This means the Apple TV and my Mac can see each other on the network.

My daughter's iPad connects to my Mac mini's iTunes library just fine, so it seems that iTunes Home Sharing is working. And when I change my iTunes library name, the Apple TV lets me know it can see the library by a error message using the new library name. There's just no handshaking going on between the two.

Can anyone think of what could cause the Apple TV to work fine one week, see all network services—even DAAP—then suddenly not be able to connect to the library?

Also: PLEASE READ MY FULL POST BEFORE FAQING ME. Please don't just post the Apple Home Sharing troubleshooting link and think you're helping. And please don't thank me for using the Apple Support Community then recommend I contact Apple for support. All other responses are very welcome.