Alastair, you may be in luck for some live jazz, since the
Hilton Hotel/LAX often features some fine talent. My wife and I had a great time there and met a couple of JC posters a couple of years ago. We didn't stay in the hotel -- only enjoyed the music.
The
Westin Los Angeles Airport Hotel also features live jazz from time-to-time, and looks like a fine choice for an overnight stay. A free airport shuttle is available. They also have a lowest price guarantee.
Of course, there's a ton of less expensive hotels available, too. A Google search will reveal countless choices.
Manhattan Beach isn't far from LAX, so it's another possibility for your stay.