Your computer has an HDMI out but because HDMI is about 7 years old system and you told your TV is 10 years old, that's not an option. The length would be no problem, my upstairs bedroom TV gets signal from my computer in downstairs study over a 15 meter (50 feet) HDMI cable. An equally long USB cable brings the mouse and keyboard receiver next to the TV so even the computer is in another room I can fully operate it from my bedside table.
Check the connections in your TV, as I said your 20 feet is not going to a problem. Ten years old means there probably is both scart and S-video? One option is a new graphics card with S-Video output which would allow you to use the TV as a second display if your TV has S-video in. S-video does not carry audio so it needs to be transferred separately from computer's audio out. You can also use a mixed cable (S-video & RCA-audio from computer to scart in TV).
A second Xbox is a good but costly alternative. Read about other options this