It seems that almost every inexpensive hotel near LAX, many with airport shuttles, have coin operated laundry facilities. Doing a
Google search for "laundry LAX airport" reveals literally dozens to choose from, including Travelodge, Ramada Inn, Extended Stay America, Four Points Sheraton, etc. You may or may not want to rent a room at any of these hotels, but I would choose one with a free shuttle and do my laundry while hanging out in the bar or restaurant. The only cost to you is a few hours of your time (which you have anyway) and a couple of dollars for laundry and detergent.
Have a fun trip!