HDMI to component converter

My stepdaughter just bought a 22in LED tv for her dorm room and wants to hook up her 360 and other devices but the tv only has one hdmi input and no component input. The problem is her 360 is an older model with no hdmi. Is there and converter I can buy that would also add more inputs? If not, what is the best device to get to convert the 360's component cables to the tv's hdmi port?

The problem is that HDMI is digital and Component is analogue so it's not just a question of an adapter cable.

The only way to do it would be an expensive converter box and I doubt they will provide a particularly good picture anyway.

Does the Screen have VGA? I remember some Xboxes could use VGA cables?