The only place I know of, if you are staying at Main Street, they have a few machines by their RV park. Your room key will get you in that laundry room.
I have stayed many times on Fremont street and have NOT found a laundry mat anywhere near Fremont street that you would dare go to or was clean enough to use. We found one on Sahara drive that we use a lot but would probably not be cost effective in a taxi(10/12 minutes by car). We found the laundry mat that we use simply by googling it. The plaza 'pre renovation used to have a small laundry mat for its patrons. why more casinos don't have one is beyond me.
24 Hr Laundromat on Maryland Parkway has pick up and delivery service. Their phone # is (702) 893-2424. You can call them and see if they are reasonable. The picture on the web looked like a clean place.