Don't know if it will fit into your schedule timing-wise, but the Four Points LAX has a day room rate of only $59 for occupancy from 9am to 6pm, so you could spend 9 hours there on that rate.

Alternately, they also have a 24 hour check-in/check-out policy on most other regular rates upon specific phone request. So you could check-in at say 5pm and have the room until 5pm the next day.

Click on "Specials" at www.fourpointslax.com for more information.

Fantastic beer bar there, BTW, and the restaurant is pretty decent. 24 hour airport shuttle. Budget Rental Car is directly across the street if you want a rental.