Last August, I went to New Zealand and I will be going back again this year. I highly recommend the trip, and would encourage you to take as much time as possible to explore.
When it comes to jet lag, I have to say, what jet lag? If you catch the late LAX to AKL flight that leaves at around 10:30 PM Pacific time, you won't have to worry about it. Think about it. You'll get airborne, and then they'll serve your your meal. By the time that is done, it will be around midnight Pacific time, so you will want to sleep. If you can't sleep on airplanes, it's a transpacific flight so you get alcohol and I've found that a nice Bailey's on the rocks helps put me to sleep.
You'll arrive in Auckland at around 5:00 AM, so it's early in the morning. If you're flying to some other place, like, for example, Queenstown, that flight will leave at around 9:00 AM so you sleep between the two flights.
Then, when you get to your final destination, you'll be ready to go.
Trust me, it really works. I got to Queenstown, took a shower, and went exploring the town.
On the way back, the same thing happened. You leave Auckland at around 9:00 PM or so and you arrive at LAX at around 1:00 PM. So, you get to sleep on the way out of there and it's right in line with your natural sleeping pattern.
In other words, there was no jet lag.