Clean the Air While You Clean the House
Your vacuum's agitator brush and exhaust whip up dust that eventually settles on the surfaces you've just cleaned. Filter out some of that dust before it settles by switching your thermostat to 'fan on.' This turns on the blower inside your furnace and filters the air even while the system isn't heating or cooling. Leave the blower on for about 15 minutes after you're done cleaning. But don't forget to switch it back to 'auto.' Most blowers aren't designed to run constantly. Plus: Here are 8 simple furnace fixes you can do yourself.
Scuff Mark Eraser
Buff Off Heavy Grime
Pick up some polishing compound at a home center or an auto parts store and use an auto buffer to polish off the offending scum. If you don't own a buffer, you can buy one for as little as $20 or borrow one from a gearhead friend. If possible, remove the doors and take them out to the garage to avoid messing up the bathroom.