The area around LAX is reasonable compared to other Los Angeles areas. It's called Westchester. Inglewood is also a good place to check for Rentals. You can join a Rental Service called Westside Rentals that may help you out. The website for that is http://www.westsiderentals.com. Craigslist Los Angeles also has listings but you have to be very careful with those.
Best of Luck to you!
I just did a quick check for you and there seems to be 90 something listings available at your price range. We would obviously need to refine this but send me an email with some more specific perimeters and I think I can help you.