As a shade tree method some people have success with kitchen foil, balled up, and water. Simichrome is one of the better chrome cleaners. Steel wool is about the worse thing you can use as it scratches the chrome. The actual chrome plating is very thin but hard. Bottom line, you can improve the finish some but once the shine starts to go there is a limit to the improvement you can make.
Carefully using Simichrome on a small mop with a Dremel if you have the patience, might give a good result. Go carefully and don't use much pressure or you'll go through the chrome layer and reveal the yellow nickel.