Harry probably does laundry-- either it got done by his Summer Court cleaning service, or Harry uses a laundromat. Washing machines are not, in general, complex and overly reliant on delicate electronics. Or, rather, they don't have to be, and those at a laundromat probably wouldn't be. Another alternative is that he could do his washing by hand. Yes, by hand-- they still sell mechanical 'washing machines' for use in a sink.