If you are looking for the cheapest option, that's washing out your items in your sink. (the shampoo/bodywash in the shower dispenser is an EXCELLENT laundry soap). After wringing out your clothes, roll them in a towel, walk on the rolled up towel to get out as much moisture as possible, then hang to dry.
If you want the ship to do your laundry, you will be able to use a wash 'n' fold laundry bag for $35. You can only send basics like underwear and t-shirts. Don't send anything you don't trust in hot water and hot dryers.
Other than that, you can use the "real" laundry, where you pay per item. I don't have the price list at hand, but I'm sure that someone will come along and post it for you.
🙂