Are there laundry facilities at The Hotel Estes?

Yes there is

There was a washer and dryer in the building near the ice machine from what I remember.

Hmmmm.... not that I am aware of, maybe contact the front desk to see?

There is a washer and dryer onsite next to ice machine. They even supplied laundry packs (Sam's Club pods) and dryer sheets! It was $1.50 to wash and $1.50 for the dryer as of August 10th, 2021.