Remove the grout with a grout rake, running the tool down each joint between the mosaic tiles. Scrub the mosaic with mild detergent, water and a stiff scrub brush. Rinse with clean water and let it dry.
Chisel out any broken tiles with a small chisel and hammer. To protect adjacent tiles, surround the broken piece with painter’s tape. In most cases, the tile will pop out rather easily. Loosen any tile adhesive with the chisel and scrub the area clean with a brush.
Apply a thin, even coating of tile adhesive to the back of a new mosaic tile and press this into the empty space. Wipe away any adhesive that comes up over the edges. Repeat this with each additional broken mosaic tile, and then let the tiles set and the adhesive dry for the length of time recommended by the manufacturer, usually 12 to 24 hours.
Mix a small batch of grout according to the manufacturer's directions, or stir up premixed grout. Load the product onto the end of a small putty knife and spread the grout over the tile, working it into the joints. Let this set for the length of time recommended by the grout manufacturer, anywhere from 20 minutes to an hour, and then wipe the face of the mosaic tiles clean with a large, damp sponge.
Let the grout cure for 24 hours and then apply a clear spray grout sealant to the entire surface. Let this dry.