Streaming from PC to TV, what is the best option for me ?

Greatings, I need some opinions on what to do, between the PC and TV there's a thin wall, I want to be able to play games on the couch with my XBox one controller and on the desk with my keyboard and mouse.

As far as I know, I can just drill a hole and use a HDMI cable. But I'm not too sure it can work properly, my monitor is only 1080p and my TV is UHD, and what about the controller ? microsoft still hasn't released the wirless adapter so I'm pretty screwed here.

Recently I heard about Nvidia Shield (still not available in my country), I understand it can stream games using Geforce experience, does it work properly with my xbox controller ?

I'd like something easy to use and reliable without the need to change settings on my PC everytime I want to play on my TV.

Best :)