Sand the plate rack lightly with a 150-grit sanding block. Wipe it down with a tack cloth to get rid of the dust from sanding.
Spray paint the plate rack with an oil-based spray paint. Move the can back and forth, evenly coating it with a light coat of paint. Select a paint color that your child genuinely loves. Allow it to dry overnight and then apply another coat, allowing it to dry for at least 12 hours.
Run a studfinder along the wall where the bookcase will hang. Mark the areas where you find studs with a piece of chalk.
Turn the plate rack facedown on a flat surface. Use a handheld drill to drill in an L-shaped bracket at the top of each vertical rail. Repeat at the bottom of each vertical rail.
Position the platerack so that the bottom shelf is just above the top of the baseboard. You want the bookcase to be as low to the ground as possible so that it's reachable for your child.
Drill in the other end of the L-shaped brackets into the wall with a handheld drill.