Mix water and baking soda to create a thick paste for a stain removal poultice.
Wet the stained areas with water and apply the paste thickly. Cover the areas in plastic wrap and leave for 24 hours. The poultice will work to leech the stains from the stone.
Rinse and wash the areas with dish soap and water. Repeat the poultice application if any signs of the stains remain.
Mix the components of the stone repair paste from the stone repair kit according to the manufacturer's instructions. Stone repair paste ingredients from different repair kit manufacturers may vary, but will likely include stone dust, pigment for color-matching, a binder and a catalyst.
Work the paste into cracks, chips and pits, using the kit's applicator tool. Allow the paste to harden for just a few minutes.
Remove any imperfections in the patched areas, using a razor blade and 100-grit wet/dry sandpaper. Be careful if you are working on a marble sink, as marble is susceptible to scratching.
Apply another layer of paste to any areas that were deep and where the paste may have sunk while drying.
Polish the areas by sanding them with progressively finer grits of wet/dry sandpaper. Start with 400-grit sandpaper and work up to 8,500-grit. Finish polishing the areas by buffing them with a soft cloth and polishing compound. Work in rapid, circular motions.
Seal the stone using stone wax or professional stone sealer. Reseal the sink periodically to retain the protective layer. How often depends on the sealer product recommendations and how often the sink is used. Wax will require more frequent applications than professional sealer.