Add option to Mirror TV screen like this:
What is it you want exactly?
You want to be able to mirror your smartphone to your TV or do you want to be able to mirror the screen from HA to your TV?
If you want to mirror the your smartphone to your TV, then HA should only provide the feature to start the cast.
Your LG WebOS TV already have screen sharing builtin, so no need for extra software like the one you show here. All that is needed is to tell the phone to cast its screen and select the device.
Is that what you want?
Or is it to cast the HA screen to the TV?
I think we must watch the video, understand it and extract the need here ourselves. 
Probably.
The download of an APK from an unknown source worries me a bit, that this might be an attempt to get people to download some malware.