When you purchased your dell did they say that your ethernet card (card that is plugged from modem to computer) was 802.1 wireless??
If not, you can go to
Best Buy, Wal-Mart, etc. and buy a wireless card. You will have to open your computer, locate an open port (slots inside the computer) take out the sliver piece in the back (or you can place the card in the same location as the ethernet card) and just snap the new card into place. The new card will probably have an antenna on the back.
If you want to bypass all of the wireless things you can always run a splitter off of the TV and just run a longer cable wire to the new desk location in the sitting room. It just depends on how much work you really want to do and if you are worried about wires being run from one location to another.