I'm in the process of putting carpet tiles in my basement. First I used a shop vac to pull up most of the major dust, then I mopped the floor with fabuloso multipurpose cleaner mixed with water ( according to directions.) I used a regular old "janitor style" mop, let it dry, and mop it again. This should do it. This should work better than a swiffer. Just make sure after you mop a little bit, dip the mop back in the bucket to clean the mop and get fresh cleaner on it. This has worked great for me. Good luck!

When sweeping, the grit is being flipped along surface of the floor to an area and then picked up in a dust pan. At work or school, if you give the floor a "white glove test" (wipe with a white rag) after a wet mopping the glove/rag will be dirty. In other words a quick slinging of a wet mop across the floor will not remove all of the dirt. If your home floor is cared for in the same manner, it will not pass the "white glove test". I love a clean floor, socks don't get dirty, bare feet do not feel gritty and dirty. Dirt on socks or feet will transfer the dirt to the chair cushions, etc. if you tuck your feet up in the chair or sofa. Think about how dirty chair/sofa cushions will become if there are children running around on the floor and then getting on the furniture. Their little legs are too short to reach the floor in order to sit like adults.

Try a solution of 1 cup white vinegar +2 cups water. Spray and wipe just like the ammonia window cleaner. The vinegar scent dissipates quickly. I've used cider vinegar, which doesn't smell quite as strong to me, but it cleans just the same. I've been using it for over 30 years and the tiles still look like new.

The only way I know to safely restore the stone is to have it refinished. You may want to check other sources for refinishing as it's only a few hours work. If you have a hard time finding a reasonably priced company, perhaps you'll have luck asking who does the work in nearby churches, hotels, hospitals etc. If their referral is a long-time in-house maintenance staff, perhaps you can negotiate with him directly as a side-job if it seems safe to do so. Some installers do this repair as well so that's another potential source instead of the franchise refinisher I think you may have gotten the quote from. You won't necessarily have to have it honed as, if it's not too far gone, they can use a solution akin to tooth polish and buff. If it needs to be refilled and honed, that's only a bit more complicated but should be doable in a day's work. You can clean it with a steam cleaner but you may wind up taking out more of the fill. Do not use any citric based solution if you try. I don't know of any good DIY treatments for bringing back the shine, but once it is polished, be sure to maintain with the right ph balanced cleaners. Maybe others will chime in here. You can also ask in the flooring and kitchen forums as some knowledgeable stone people frequent there. Good luck!