Try to get a day room at an airport hotel. L.A. traffic can be horrendous. Although the hours you'll be needing a room doesn't fit the typical "day room" hours, nothing to lose by asking.

Or, take the Hilton Airport shuttle and pay to use their fitness facilities. You can work out and take a shower, eat something and have the shuttle drop you off at TBIT.

If you are intent on staying at the airport, you might consider purchasing a refundable DL ticket, go to the DL CRC in Terminal 5 via the underground tunnel from Terminal 6, which is connected by walkway to Terminal 7 where Ted flights usually arrive, which keeps you airside the whole time. At the CRC, show them your ticket and pay $25. You can shower there (shampoo, shower gel, towels, hairdryer - toothbrush kits, shaving kits, brushes and deodorant upon request; have free drinks then walk/stagger to TBIT. CRC stays open until 2330.