Get a spray bottle and test out your laundry detergent (I like liquid better, but can use powder, or even handwashing liquid like Woolite). You should be able to water it down a little and check it under black light to see if it has phosphorous (most of them do). Do about 1-2 ounces to 16-20 ounces of water. Turn on the blacklight and start spraying down the costume. Allow to air dry and it should be fine.
Or you could get a packet of Rit whitener, but most folks already have laundry detergent in their house, and a spray bottle mixture is the easiest way to apply.