You can install any VNC server app on your Android, such as VMLite VNC Server. On the RPi you can use a VNC viewer, such as SSVNC (sudo apt-get install ssvnc) or xtightvncviewer (sudo apt-get install xtightvncviewer) to connect to your Android VNC server and take control of your Android desktop. This does require you to have access to the RPi - so it doesn't work like a typical "cast" solution.
If casting is what you want - I'd just fork out the extra $35 and get a Chromecast - I'm using one regularly plugged in to a projector during classes - my Android tablet just does a Screencast from the Settings menu and performance is fantastic. There are "casting" solutions for the RPi (check out PiCAST or the suggestions in this thread on raspberrypi.org) but they typically only do remote invocation of content such as YouTube videos and the like. No screen mirroring.
You might want to look into writing a clever script on the RPi which you can call from a webpage, which in turn launches a VNC viewer on the RPi using the VNC server IP of your Android. That way you can just visit the webpage on your Android and have the Android screen "magically" appear on your TV via the RPi. When I say "clever", I mean that you could make the webpage use the IP address of the Android visiting the page as the address of the VNC server. That way your webpage would just have 1 big button saying "Cast my screen" and once pressed, the webpage initiates a VNC viewing session to the IP of the connected Android device.
