I've used the facility next to L'Absinthe right at Passage de Jacobins. It's easy to find and is self-service. I was the only one there when I was using it. They have a few larger machines (you pay extra for) that did the job nicely. The smaller machines were almost uselessly small. I brought my own soap - and unfortunately I don't remember if there was soap available at the location.

Easy insurance policy is to just buy those little dual use washer/dryer sheets and bring 'em with you from home. They take up virtually no space in your bag and will be cheaper and probably higher quality than buying soap on the spot.

The Westin Concierge told me about it, that's how I found it in the first place. Worked out just fine.