I see 3 solutions here.
1. Buy a new router and set it for repeater mode (wireless connection between routers) so that it connects to your old wireless and extends the signal.
2. Buy a new router and set it to access point mode (wired connection between routers). Place this in the adjacent room and connect it to the existing "lan cable". Have it broadcast its wifi to the room that needs it.
3. Run a long Ethernet cable to the room that needs internet from your current router to the room that needs it, routing through walls as needed. (Wall plate would probably look prettier)
Cant do the third, the room where I need the connection is Soundproofed, So drilling any new holes might introduce unwanted sound.
I had thought about the second one as well, but how much lag do you think this would cause while using Steams In Home Streaming?