Maybe get a cheapish hotel for the night near the airport in KOA. Just glanced on Kayak, there are a couple of guest houses near that are under $100.

In LAX you should have access to the Flagship Lounge which has nice showers. I'm not following what you mean by you have to commute to a different city in the US. But as long as your status isn't from AA or AS then Emerald status will get you and a guest into the Flagship lounge (or Admirals Club, AS longe, or either QF lounge if they are open, but FL is definitely more convenient than the QF lounges).