Open the closet door and brace it from behind with your foot.
Work from the back of the door and place a narrow-blade wood chisel at the top of the slot that the broken slat slides into. Gently tap the wood chisel with a mallet to cut the slot open all the way. Repeat the process on the other end of the louver.
Remove the broken slat by tapping it with a screwdriver and mallet from the front side of the door.
Take the broken louver slat to your local home improvement center and match it up with a wood lattice strip of the same width and thickness. Lattice boards are only available in certain pre-cut lengths. You will need to cut the lattice board at home to custom-fit it to the louvered door.
Match the color of the broken lattice to a can of spray paint of the same color.
Use a jigsaw to cut the lattice strip to the correct length and sand it with coarse-grain sandpaper to smooth the cut edge. Spray-paint the lattice board and wait for it to dry.
Slide the lattice strip into the slots in the door where the broken louver was removed.
Drill a small diameter hole into the door frame behind the lattice strip. Gently tap a small nail into the hole to form a barrier to hold the replacement slat in place.