The budget plan:
1. Go to any airport hotel. As previously posted, the Hilton is closest.
2. Take the elevator to any guest floor.
3. Look for the housekeeping cart. Go into a room that the housekeeper has just started cleaning.
4. Say "I'm sorry, our flight was delayed and there are a few more things we'd like to do here before we leave. Can you work on another room for a while? We'll be out in an hour or so."
4a. If it doesn't work, try a different floor.
5. Unless you were unfortunate enough to pick a room previously occupied by a large group or an exceptionally messy person, there will be unused towels. If there are no unused towels, or the room is unacceptable for any other reason, go to step (4a) above. Repeating the process on the same floor, especially with the same housekeeper, is not recommended.
6. Shower.
7. Leave.
(Truth in advertising: I have not tried this. If you do, please post what happens.)