Sorry, no laundromats on the Zuiderdam - it's a Vista ship.
They will offer laundry/drycleaning/pressing packages which we have used and have been very pleased.
Even when we were on the other HAL ships that had laundromats, I never used them. I'm on holidays - I do enough laundry at home.
I do wash out my delicates in the sink and then just hand them on the laundry line over the bath tub - it's a retractable line, in case you can't find it.
Enjoy your cruise.