If you don't want to leave the airport, buy a refundable ticket on AC (or NW, AF or NZ); these all leave from T2. The NZ lounge in T2 is the best *A lounge at LAX, possibly the whole country, and they even have the generosity of slapping a *G designation on it. Fantastic food and drinks, nice showers, and a good experience.
If 7 hours in a lounge doesn't appeal, I would suggest heading to In and Out when you land. Around that time, theres a good bank of heavies to Europe and Asia, and some Asia inbounds as well. Burger and shake with some Jet A mixed in, then head to the NZ lounge to enjoy.
You could also have a shower at a hotel or gym and a nice dinner in Manhattan Beach or Santa Monica.
My $0.02