Close doors, open windows and lay down a canvas drop cloth to eliminate the amount of dust that travels throughout the rest of the house during your grout repair. Place masking tape along the edges of the tiles, leaving the grout exposed. Put on protective eye goggles, work gloves and a breathing filter before you begin the grout-removal process.
Use a grout saw to cut along the grooves between the tiles. Work the saw back and forth until you have removed all the grout, and there is a clear path to the surface beneath. Use a hammer and chisel to remove any chunks of old grout that will not let go. Be gentle so you don't chip the tiles.
Use tile cleaner, clean rags and a brush to thoroughly clean the tile and the channels between them in preparation for applying new grout. Dirt and dust will affect the grout's ability to adhere to the surface and to the tile.
Squeeze a bead of silicon latex caulk into the space between the tiles. The bead should be large enough to fill the bottom of the crevice while leaving the top 50 percent open for grout application. Remove the masking tape from the tiles.
Apply your tile grout with polymer additives to the tile by spreading it over the surface using your rubber-coated trowel. Press the grout down into the joints, and wipe off all excess with the side of the trowel held at about a 90-degree angle.
Wet a large sponge and run it over the surface of the tiles to remove all grout residue and grit. If you come upon a spot with too much grout, remove it by rubbing it down with your sponge. If you come across a spot with too little, use you hands to fill in the space. Continue to mop and clean the tile surface with the sponge, dipping it into clean water frequently so grit does not lodge in the sponge and dull the finish of the tiles.
Polish the surface of the tiles by rubbing a towel in a circular motion similar to that used when waxing a car. Allow the grout to dry according to the manufacturer specifications (typically from 24 to 48 hours). Apply tile sealer according to manufacturer specifications, and allow it to dry and set.