Drill a series of diagonal holes across the broken tile with a ¼-inch masonry drill bit. Space the holes approximately 1 inch apart.
Insert a chisel into one of the holes and gently strike it with a hammer to loosen the tile pieces. Continue to tap along the drilled holes in order to free the tile. Remove the pieces. Scrape away excess tile pieces along with the existing grout with the chisel.
Remove the mortar from the basement floor beneath the tile by using a stiff-bladed scraper. Get rid of enough to make the surface smooth.
Mix a small batch of latex-fortified thinset mortar in a container according to the product instructions. Spread a layer of the mortar onto the area where you’re replacing the tile. Make the layer approximately ¼ inch thick, using a notched trowel to apply it.
Place a new drainage tile into the spot where you removed the previous one, pushing the new tile into the mortar. Allow the mortar to cure overnight.
Apply grout between the replaced tile and the original tiles, pushing the grout between the joints using a rubber float. Wipe the surface with a damp cloth after the grout cures.