To put it simply, a switch directs traffic. You can hardwire the traffic with a single cable connecting the homerun cables from the router room to the tv room. You can do this with only a laptop and no special tools needed. Just plug an ethernet cable from one of the (most likely) 4 outputs into the rj-45 wall jack going to the patch panel. Then take your laptop to the patch panel, and test each port of the patch panel by plugging another cable into the LAN port of the laptop. You've found the right one when you have a network connection on the laptop. The ethernet cable will remain plugged into that port of the patch panel, and you will do a similar process to find the homerun cable going to the tv room. This will involve a little bit of running around. You will try plugging the other end of the cable at the homerun into each remaining port of the patch panel. Each time, run over to the laptop which is new connected to the rj-45 jack in the tv room, and see if you have a network connection. Once you do, you're done.
It's easier with testing devices, but there's no sense in buying them if you can do it without in less time than it takes to go to the store. The advantage of testing tools is they will tell you if the pairs are terminated properly. If they aren't it might still work, but you won't know until you try.