Yeah I'm not up to hang out in the middle of night at LAX. sounds so unsafe.
And I just saw a consolidated LAX layover thread. Sorry I didn't notice it before, but it looks like it's more for international travellers and have suggestions to go to culver city, In-N-Out, etc.. Didn't seem to apply to me.

Anyway, it looks like Ted's suggestion sounds the best. Not a member of 24hr fitness. But midnight to 4am at Hilton cafe and then take their airport shuttle to LAX for my 6am flight doesn't sound like too bad of a plan......