For anyone sleeping over in LAX, I recommend TBIT. It's a bit of a haul from T6 (20 minutes or so, felt longer after leaving the lounge), but there are a few spots where you can stretch out. There are some couches in the area behind the stores on the left side of the shopping area. We stayed behind the Tumi store until the cleaning crew came. Next we went up to the 5th floor and were in the bar area of PF Chang's. Some airport employees also used the space to hang out for an hour or two, so it appears to be a known quiet area.

It was great meeting other VX enthusiasts, where I didn't have to explain why I was taking this particular flight.