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Pour water onto the surface of the slate. Take an oak stick about ¼ to 3/8 inch thick and 1 inch wide cut square at the tip and scrape at the grout with the cut edge. Use the stick to remove as much of the grout as possible, concentrating on large clumps of the material. Keep the surface of the tile lubricated as you chip away at the grout with the wood. This should chip away standing grout pieces, leaving you with only a thin covering of grout haze. Rinse the surface with water to remove any particles of grout left on the tiles.
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Wet the surface again and scour the surface with a nylon scouring pad. Keep the tiles wet as you scrub the surface to remove as much of the grout haze as possible. Rinse the surface of the tile and pat it dry to determine the rate of removal. Repeat the process until you've removed as much of the haze as possible, leaving a thin film of grout.
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Remove the grout film with a mixture of 1 cup of sugar dissolved in 1 gallon of water. Spread the mix over the tiles containing grout and keep them wet for two hours, reapplying the mixture as necessary. Scrub the tiles with the nylon pad to remove the film. Scrub until dry and examine for progress; repeat as necessary to remove the remainder of the grout film from the surface. Rinse the scouring pad in clean water while scrubbing the floor to keep from spreading loosened grout residue.
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Clean the surface of the tiles with a cleanser formulated for slate and seal the surface with a layer of grout and tile sealant, using a brush.
Check out this related video from Homesteady on Youtube.