If your device isn't on the (relatively) short list of devices that will support the new Screen Cast option for Chromecast, and it's rooted, then XDA Developer r3pwn has an apk for you.

Head on over to XDA (and click the thanks button if it works). Note: If you don't already have it, you will also be prompted to install sqlite3, but the process is automated so the only issue with installing it would be whether or not you trust them.

I tried it out on my Nexus 7 (2012 WiFi) and it worked pretty well. There's a little delay in switching to landscape mode, and if there's a lot going on, there's a bit of lag. I tried playing the Twitch TV app, and it casts, but there was a slight delay in the audio starting, and a few moments of lag here and there.

And if you experience any issues, you can help out by noting it in the comments.