I bought a Shark steam mop and I'm sold. I think the Shark could conquer the dirt and grease, but it would take more than once. Nice thing about the Shark is that you do it all from a standing position. Pads are washable and reusable, but I'd get at least two.

This may sound unorthodox, but have you tried Scrubbing Bubbles & a mop? SB works great on our floors, one is Flexitec vinyl & one is just stick on tile, hasn't hurt either. I actually get on my hands & knees to do this, but since you can't, I would suspect a mop would work too. You do have to rinse w/ clean water afterward, but it will be worth the effort. You may also look into some kind of degreaser... Not sure of the effect of this on flooring though.

Hope this helps,
Mc_Hudd :)

Try Krud Cutter from Home Depot. Nothing works better I promise!