You cannot have two Modem/Routers connected simultaneously to an ADSL enabled phone line.
One option might be to use Homeplugs. Plug one into a mains socket in the study and connect that by ethernet cable to your router. In the lounge plug in a second Homeplug and connect the TV to that. If you need *two* ethernet ports in the lounge, you'd then also need a Network Switch.
>>> ...or you have to buy an expensive wireless dongle
Wireless dongles are not necessarily expensive, e.g. this 150Mbps 802.11n adapter under £10 click here
Should I upgrade to Windows 10? 8 reasons why you should upgrade to Windows 10... and 2 why you…