refinishing old slate floor

We just bought a house with slate floor tiling in the kitchen, hall and bathroom. Replacing the floor is not an option financially. It has a hideous coating of what I am assuming is years of wax/dirt buildup - yellowish, browninsh and black. I tried to strip a small area with ammonia and it removed a lot of the shiny wax but not the crusty yellow bottom layer that seems to be stuck in the grooves and pits. I spent like an hour on one tile, and no amount of elbow grease affected it.  I tried a heavy-duty stripper from Home Depot with no results at all. The stripper at the janitorial supply store was too toxic in my opinion - I would like to do this myself but I don't want to actually die in the process. My husband thought we should sand it down but that seems like kind of a crazy idea - you would have to smooth it all out to one level to get the wax out of the grooves. Any suggestions??? Thanks, Tina