There's no need to simply live with a broken laminate plank. There are two methods for fixing the problem. The first involves removing planks until you reach the damaged one. This method works best for broken planks that are near a wall and that are a part of a floor not glued together -- also referred to as a "floating" laminate floor. The second involves removing the damaged plank using a circular saw. This method works best for planks in the middle of a floating laminate floor or flooring that has been glued at the planks' tongues and grooves.
Disassemble the Flooring
Remove quarter round molding and baseboards with a small pry bar and thin putty knife or thin piece of scrap wood. Use a utility knife to break the seal between the wall and the baseboard. Use the putty knife or thin piece of scrap wood to protect the wall or floor as you pry with the small pry bar. Mark or number the pieces you remove so you know where to put them back. Pull out any finishing nails that remain in the floor with a pair of pliers, noting their length.
Lift the corner plank to disengage the interlocking tongue and groove. Continue to lift, unlock and remove planks until you reach the damaged plank. Keep the planks in order or number them so you can return them to their original positions.
Remove the damaged plank. Replace with a new plank of the same length.
Return the planks you removed in Step 2 to their original positions and reattach the quarter round and baseboards. You should be able to reuse the finish nails that remain in the quarter round and baseboards. If you need to use new finishing nails, use the same length as the existing nails. They'll likely be 1 to 2 inches long.
Cut Out Broken Plank
Drill holes using a 1/2-inch drill bit that are 1/2 inch from each corner of the broken plank at the top and bottom, and 1/2 inch from the sides of the plank. Holes down the sides of the plank should be drilled 4 to 5 inches apart. The holes will be drilled around the perimeter of the plank. Be careful to drill just the depth of the plank -- you don't want to drill into the floor beneath the laminate. Wrap masking tape around the drill bit to mark how deeply you can drill.
Connect the drilled holes around the perimeter of the plank using a pencil and a straightedge.
Cut through the plank along the lines you marked in Step 2 with a circular saw. Set the saw's cutting depth to the thickness of the plank. Start cutting at one of the holes you drilled.
Lift the center of the cut plank with a small pry bar. If it's a floating laminate floor, lift up on what's left of the plank to disconnect the remaining pieces from the adjacent planks. If the flooring has been glued at the tongue and groove, clean the edges using a chisel. Remove the glue using a tongue and groove cleaner tool or the tongue edge of a small piece of laminate. Clean up any sawdust or debris.
Measure the newly created space and use the same size replacement plank. With a utility knife, remove the bottom of the groove on the end and on the side. Remove the tongue from the end of the plank.
Test fit the plank. Check for high edges. If necessary, adjust the fit by sanding the replacement plank along the sides or bottom. Use 120-grit sandpaper.
Place a small amount of glue in the tongue of the replacement plank and the groove of the adjacent plank on the floor. Put the new plank in place and wipe away any excess glue.
Place heavy objects along the entire plank and let the glue set for 12 hours before walking on it.