Yeah me, I'm going to Buenos Aires!

A few details to work out - would love some guidance from knowledgeable FTers...
6am UA flight from LAX on Tues, need to check-in at 4am. Since there aren't any flights from Phx arriving at 4am, looks like I'll be arriving 10:30pm Monday night. Any decent spots to sleep at LAX? Per sleepinginairports.com, LAX gets a 'poopy' rating.

No sense in getting a hotel, I'd only be there for a few hours and fret over getting back. Although I'm a budget traveler, I'd happily pay for a lounge day-pass but it closes at 11pm. Any advice on good snoozing spots, ways to 5.5hrs?
Also, I've got 7+ hrs at IAD. The Nat'l Air & Space Museum, which is close by, was recommended as an idea. Any other suggestions?
(now must plot who to borrow ambien from for my 10.5 hr flight IAD-EZE...)
Thanks!