There are definitely at least two laundry shops on Soi 1; I noticed them today while riding a motorcy taxi down Soi 1.
You could take a motorcy taxi from in front of the JW, go down Soi 2, cut through Raja, right on Soi 4 to the Wan Laundry, opposite the Omni tower, drop off your laundry and take the same motor bike back for ~ 50 baht. Easy peasy.