This is the Displays settings window, this is where you turn displays on/off, and where you choose display options. If you have your TV plugged into your notebook, your notebook should be able to see your TV.
Your laptop is the box in RED. See the switch that shows ON? That means your display is enabled. Now, if you click on the GREEN box, which should be your TV display, does that switch say ON as well? Now I remember you said the mirror box was grayed out, I just wanted to make sure the display was at least enabled, hence switched on in Displays options.
Now, here's the thing, your notebook is using a common resolution of 1366x768 for a small 14" screen. That however, is not a common resolution for a TV, common resolutions for TV's are 1280 x 720, 1920 x 1080, 3850 x 2160. In order to mirror the TV, it has to support that lower resolution, and I doubt that it does.
This could also be the reason why the box is grayed out. Which means, you either have the option to extend the desktop to your TV instead, or, switch to using the TV as a main display only. Which is what I am doing BTW, I don't use my notebook screen.