For your "Carborundum", try "silicon carbide". The black stuff commonly sold as wet and dry. It's harder than aluminium oxide and more appropriate to stonework.
One of your biggest problems will be dust. Unlike wood, stone dust doesn't compress and so the paper will tend to clog. Keep a vacuum running as a dust extractor, just to keep the abrasive cutting. Working it sopping wet will help too.
For a quick fix from the local DIY shed, try the bright orange Plasplugs tile-fitting tools and get a "tile file" - a grit-covered latticework. This is a carbide abrasive on an openwork substrate and good for working dusty stone.
What colour is the slate ? The Chinese stuff is quite soft (but variable - red / brown is softer), fresh Welsh grey / purple is hard, and green Lakeland slate is very tough going.