The sink and /or bucket are your best buddies here. It's certainly the least expensive option for delicate and unmentionable things - as long as you stay on top of it, and don't plan on doing a "week's load" in a single night.
Buy detergent and bucket at the local "Wal-Mart" here, called "HyperCity." Familiar brands are available. Use detergent sparingly and leave plenty of time for drying!
Danny found a Laundromat about twenty minutes drive from the hotel and had good success there. Our driver "has a friend" who does washing. A couple of our group are "giving him the business." See what I did there? The rates seem very reasonable, especially given the convenience.
Our hotel has daily pick up and delivery for dry cleaning and a laundry service. It's on the expensive side, as things go in India. Regarding dry cleaning, we did notice several web site reviews that reported occasional dissatisfaction with treatment of delicate items. So far, though, none of us have experienced any problems.