There's not much more economical than the hotel sink, and travel laundry packets, and a travel sink stopper (in case your sink doesn't have a good-enough one).

Now, it won't work well for bulkier clothes, but for a couple T-shirts it might be practical. But for drying time, it's easier to do (at least a full day before check-out) on a multi-night stay, than trying to fit both washing and drying time into a one-night stay.

But that's the same with most hotel laundry services (that are not coin-operated washers/dryers): They only tend to be "leave it in the morning and get it back later in the day", which again doesn't work on one-night stays.

So the AC wouldn't work anyway, given that you're staying there only one night, unless you were able to find a coin-operated laundry nearby.

FYI: In some cities in the world, there are laundromats aplenty, if there are a lot of people living nearby in small apartments with no washer/dryers. On the other hand, in places where most everyone nearby lives in single-family homes, there tend to do no laundromats except perhaps a washer and a dryer in a campground nearby (if there are any campgrounds nearby).