Remove the broken piece or pieces of slate. Chisel out the grout surrounding the offending piece. Use light taps of a hammer to avoid accidentally damaging surrounding pieces. After you have removed the grout, insert the chisel under the edge of the piece, and use the hammer to chisel away the piece of slate. Scrape the surface clean with a floor scraper to prep it for installation of the new piece.
Sweep the area clean of debris. Spread an even layer of latex modified thinset mortar onto the area where you are installing the new piece of slate, using a notched trowel. Press the slate tile firmly in place. Apply pressure as you gently move the piece in all directions. Listen for air being pressed out from behind the tile. When no more air comes out, the bond is complete, and you can let the piece dry in place. Allow 24 hours before grouting.
Force grout into the joints around the piece of slate tile with the grout float, and clean any excess off the face of the slate, using a damp sponge. Allow about 20 minutes to elapse, and then use the damp sponge to gently wash the grout off the tile, and smooth the joint. Avoid excess water, and use circular motions. Dip the sponge in clean water, and wring it out as needed.
Wait another 30 minutes, and then wash the slate tile again with clean water to remove the final residue. Use damp sponge, but this time pull the sponge in a diagonal motion across the grout joint. Do not wash the joint, as this will pull the grout out of the joint. Allow a minimum of 24 hours for foot traffic.