Oh man... fun topic.. if it were 2002!!! I used to have to do this all the time to get secure extractions to work on scratched discs.
My recommendation... Brasso! Get yourself a bottle of Brasso for $3 and soft, lint-free cloth and you will be in business. Polish from the center outwards and it fix all but the most severe of scratched surfaces.
Both the home-kits (like what you've purchased) and the high-tech machines H2O refers to will work, but hardly as cost-effective. Just make sure you keep things fairly clean when using brasso... it's not very viscous and will drip a bit if you use too much, so just have an area set aside for working with it.