TVs each have a proprietary mirroring method, and unless the TV and the phone are the same make, they usually don't work together. (It's surprising that your Samsung works on your LG TV. Mine doesn't see my Vizio.)
As for Chromecast, the app works n the N4, N5, N6 and N7 (2013), the 9 and the 10. Just install
Chromecast, plug the Chromecast device into a spare HDMI input on the TV, power the device (either from a USB port on the TV or with the supplied wall charger), make sure wifi is available (the device will update itself - it takes about an hour of watching nothing before it's ready to go) Then run the app. (See
Chromecast mirroring for instructions).
It mirrors the phone screen to the TV, so anything you see on the phone (even if the screen blanks), you see on the TV - except where your finger is, so you have to look down at the Nexus to tap anything. (And holding the phone in portrait mode will seem weird on the TV.)