Getting internet to sleepout

Just after some help with setting up internet in a sleepout that is about 20 from the modem router. So far after doing some research I am thinking a 30 m cat6 cable outdoor rated in polythene piping under house along fence then through window. Then do I attach it to a switch box with a wireless thing plugged in (need suggestions on what to get here). I have tried power over internet adapters and would need to install another plug on same circuit by modem to make that work, I am also not sure they would have enough strength to stream Netflix etc

Aim to have internet out there for Netflix plus wireless for laptop or mobile and Budget is $100-200

There is a tv cable underground to the sleepout, builders that built the place 6 years ago said it had all the cables needed but can only see phone jack and tv arial points.

All help especially where to get cheap bits most appreciated