Connecting router to extender?

I'll try to not make this long-winded. I live in a studio on the same property of a main house that's about 200 feet away. Main house has WiFi and I have decent signal in the studio. I just bought a Phillips Hue second gen base and two bulbs and it requires ethernet connection. I'd like to capture the WiFi and somehow convert it to Lan. My idea is to buy a WiFi extender (Netgear WN3000RP) with an ethernet port and connect that to either a plain ol' router with 1 input and 4 outputs or possibly buy a wireless router with multiple ethernet outputs to connect Hue, Foscam, etc. Am I onto something or totally wrong? Side note, there is a wall plate in the studio with ethernet port that says "cat5e", two phone jacks and two coax connections all on the same wall plate. Could this possibly be the answer? How do I check if there's signal in either cat5e or coax? Landlord is totally oblivious about this stuff and I'm not much better. Any help is appreciated!