Surprisingly Seattle Children's Museum DOES NOT have a nursing room. Sad for a place that's solely dedicated to children and parents.
On the positive side, the state of Washington protects a mother's right to openly breastfeed whenever and wherever she is allowed to be public or private. So breastfeed right there if you are comfy with that.
Note: Report mom-shaming or harassment while nursing in public or near lactation rooms (breastfeeding rooms), at Moms Pump Here mobile site https://www.momspumphere.com