Regular dish sponges can easily be cleaned in the dishwasher along with the rest of your dirty dishes. You can place it on the top rack standing between the pegs, so it won’t be moving around during the washing process.
Obviously, the entire broom won’t be able to fit inside the dishwasher, but the bristle head can easily be placed in there. Just make sure to put them upside down, so they don’t get damaged.
Kids’toys get dirty very fast, and the easiest and fastest way to wash them is to place them in the dishwasher along with the dishes.