Rotorua!
Now that I can do...
(Lived there for 11 years!)
Aheeeerm, well yes you could very well stay the night there. It's a very touristy place so they have quite a few nice places to stay.
Once there you should prooobably take a look around all, you know, the thermal areas.
Also Skyline Skyrides - Luge, gondola, activity centre, and also restaurant is a must-visit really...
The Polynesian Spa is quite nice and relaxing too.
Now, you say you're staying 3 nights in Hamilton?
That's fine, I suppose. Hamilton is a good base. From there you can check out Cambridge, Matamata (Hobbiton) and relatively small towns such as Tirau. You could also check out the place that the Maori Queen lives in, which is near there too.
Now, I'd suggest going through the desert road through the middle of the country if possible so you can see Mt. Ruapehu and.. the neighbours.
I don't know much of Taranaki or East Coast, so I can't help you there too much.
But: Is the Sky Tower in Auckland on your itinerary. If not, well, it should be. One of NZs greatest landmarks. Certainly the tallest
.
And you should also check out Rotorua Museum, whilst your there.
If you have a week, considering you can get anywhere you want to in the North Island within 24 hours straight driving, I'd just try to experience as many cities and towns as possible.
And we drive on the left side of the road. And the steering wheel is on the right, yes.