"Err," Nadia pondered, but borrowing another wand to perform Scourgify, or complicatedly perform the spell to a mirror didn't look like t he things this guy would like as answer. So...

"There's kits for cleaning brooms and wands both, with instructions as to how to do it the proper way," she said raising her hand. "And with tools, magical, to aid the cleaning better. Like a cloth that naver gets dirty, to polish the wood. And. You might awnt to start with the tip downer, and never point the wand at yourself."