He probably uses a dry-cleaner. Any smart person knows you can't machine wash things like the 10th Doctor's coat, 4th's scarves, and the ties.
It's also possible that Rose just hasn't found the washing facilities. In fact, we never see her venture beyond the control room during her time on the series.
If she'd asked the Doctor, he probably would have told her. But I suspect 9 gets some enjoyment out of the fact that Jackie's being made to do Rose's laundry. And, contrarily, he wouldn't really enjoy seeing Rose be unnecessarily "domestic".