Is this TV a LCD/Plasma/Thin? Or is it a CRT? If it is a CRT TV, then you should really consider saving for a new TV. If your current TV is a LCD/LED/Plasma without an HDMI port, then an HDMI to component converter would make more sense.
The reason that cable you bought didn't work is because component and HDMI are nothing alike and a converter box with "smart" circuitry and chips is needed. HDMI is digital and data can be streamed in different formats. Component is analog and only has one format:analog video.
Well, here is a cheap made-in-china HDMI-to-component converter box.
http://www.monoprice.com/Product?c_id=101&cp_id=10114&cs_id=1011406&p_id=5971&seq=1&format=2
However, I heard that anything that is HDCP copy protected is...