E29 is right, these tough plastics clean easily. Years ago, when my kids were small, I got into using "Charlie's Soap" because it is clean and pure with nothing extraneous in it, and I have never quit. It cleans well and leaves no scent, my go-to laundry mix is 4 parts Charlie's and 1 part Biz (or other oxi-product) and I use it for key caps because it rinses well.
I do think hot water is a good idea because a lot of the dirt is the direct or indirect result of skin or food oils, and oils always soften under heat. And I have found that 10-20 minutes is as good as 10-20 hours.