Ordeal depends on class of travel. In BP it is fine, you'll have slept AKL-LAX, be fairly fresh at LAX, off the plane early, just go to the immigration queue for processing then be directed to the transit desk before re-entering the terminal for priority security and into the lounge for a shower (which always is a good idea. If you can take a change of some clothes, such as underwear, shirt, it can help a lot too). PE will be tolerable, but the queue will be longer. Still given, usually, a majority of NZ2 passengers aren't flying on to LHR on the same flight, you should have enough time to get through to the lounge. Elite status will get you access to the priority security queue too. In economy it IS an ordeal, and depending on queues you may not have much time in the lounge when through.
Once you get to the lounge, just ask when NZ2 boarding is expected to be announced, as you'll then know what time you may have to shower etc before having to go to the gate. It's up to you if you like boarding early or not, but if in economy you may want to board early if you want to ensure overhead bin space above your seat for hand luggage.
Basic procedure is:
- Exit aircraft, get handed transit card and directed to immigration.
- Undertake electronic processing of entry with your passport to verify your ESTA.
- Queue for checking with immigration official
- Directed to separate check for transit passengers for NZ2 to re-enter departure concourse landside
- Go upstairs to security (use priority if class of travel or status allows)
- Enter departures airside, with option to go in lift to Star Alliance lounge.
- Remain in lounge until flight called to go to gate (or go earlier at your leisure).