Cut a 12-inch long piece of solid hardwood, such as maple or oak, 3 inches high and 3 inches wide on a table saw. Cut this piece to a length of 4 1/2 inches on a miter saw.
Use a tape measure to mark the center of one of the 3-by-4 1/2-inch sides of the block. Place the block into a bench vice with the marked face up, so that there is at least 1 inch of space between the bottom of the block and the vice. Position the tip of a 1-inch spade bit on the mark and bore through the block of wood until the bit comes out the other side.
Use a 3 lb. hammer to gently tap a wooden hammer handle into the hole in the wood block. Tap it in until the end of the handle is flush with the bottom of the block as it sets in the vice. Loosen the vice and turn the mallet so that the handle is parallel with the vice's jaws and the top of the handle is exposed. Tighten the vice.
Drive the metal wedge that came with your hammer handle into the split in the center of the handle's top to spread the top of the handle, to lodge it in place in the mallet head. Reposition the mallet in the vice, so that the vice is gripping the handle and the head is up, with the top of the head flat.
Fill the area around the top of the handle with liquid hyde glue, available in hardware stores. Allow the glue to harden. Sand the head of your mallet smooth and round the corners over slightly with 100-grit sandpaper.