Carefully line up the drawer with the front edge of the box, and then glue some wooden stops behind to prevent it from slipping too far inside. Now mark the drawer sides where the magnet slots are, and cut a matching recess in the drawer sides. The bar magnet should be able to slide freely into and out of the slot in the drawer.

Place the bar magnets into the slots, and temporarily place the MDF top onto the box. You can test that the magnetic locks will work by passing a magnet over the area of the locks, sliding the bar magnets back and forth in their slots.

Remove the MDF top and glue the tiles to it using contact adhesive. Trim the edge of the box using hardwood strips. Make sure that on the front side you glue the trim to the drawer only, not to the box sides. The trim should be wide enough to cover the box and the edge of the tiles. Finish the corners with a wooden corner molding of your choice, but remember to glue it in such a way that the operation of the drawer is not affected.