Will a router work if it is not connected to the Modem?

What you can do is to purchase a wireless receiver and connect it to your TV via the ethernet port on it. You will also need a wireless router attached to your modem so it can transmit the signal wirelessly to the receiver, along with any other wireless devices you may purchase in the future.

"What exactly do you mean by your TV can have internet access but doesn't have wireless capability built in?"

I think what he means is that, like my LG TV, it has Wi-fi capability, but I had to buy LG's USB Wi-fi network adapter which plugs into the USB port of the television and allows the TV to receive the wireless signal from the wireless router.

What he needs is the modem, a wireless router, the USB Wi-fi adapter, and to properly configure the television to receive wireless signals (e.g., entering the WPA in the TV's network settings screen). Most TV's with Wi-fi have an option to "automatically detect" the router's settings so the user does not have to manually enter them. This typically requires the user to temporarily disable the firewall.

What exactly do you mean by your TV can have internet access but doesn't have wireless capability built in? You saying it has an ethernet port? Anyway a wireless router will probably work (except in this usage it will probably acting as a repeater or bridge)

YES But you wont be able to go online but you well be able to share files and printers ext using the routers D H C P server and connect to your network with wireless enabled type 192.168.1.1 in you address bar to assess the routers configuration panel

You need a wireless router. Simple as that.

Buy a 25ft ethernet cable, drill a hole in your cieling, and connect the etherent cable to your tv and the other end in your modem. if you dont want to you that, buy a 50 ft cable and just wire it along the wall.