P.W.
I sell a non-toxic cleaner that is a fabulous product. I use it on my Mexican tile, pergo, wood, silestone counters. I would think it work great on slate too as long as it IS SEALED. It works on stone, tile, clay, granite if sealed. This stuff is amazing on more than all this. I use it to clean just about everything! And it comes concentrated so it lasts forever.
Take a look:
http://www.shaklee.net/healinghappens/product/00015
P.