Mopping a slate floor



What is the best way to mop a slate floor? What cleaning solution should we use? We have slate tiles in the kitchen with concrete mortar. You can see from the pictures that the surfaces of the slate tiles are uneven. They are also porous (water dripped on them is absorbed). I sweep regularly, which isn't easy because of the uneven surfaces. I have mopped once with a Swiffer Wet Jet but I'm not sure whether that's OK to use repeatedly.



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I used dawn and water to clean my slate floors, and then steam mopped them. They need to be sealed as well, but there are a few stains that I can't get out....possibly oil? I don't think previous owners sealed them, and they're in the kitchen. :(
Any suggestions?