I think there may be a small laundry shop on Soi 1 on the right side? BTW, a great lunch-time food-court is a bit further down that soi on the left.

There may also be something in Ploen Chit Center, the "twin" building opposite the hotel on Soi 2. Look in the basement level, which is where the services usually are. (Also a great food-court there on floor 7.)

There is a laundry shop on Soi 4 down a bit, opposite the Omni Tower, and there are probably a ton of shops in the Soi 3/5 area.