You could do S and N in 21 days, fairly okay if you do not do lingering things (long treks,etc).
We did Melboure for 6 days then N island in 14 days. 5 in Auckland and 9 driving around N Island. Our first stop was Te Kuiti as it allowed a stop from Auckland for the Hamilton Zoo then a motel with 3 washers and dryers, as we packed for only 11 days. Then spend one night in New Plymouth. Two nights in Wellington (Te Papa museum is a must). Two nights in Napier and one in Rotorua.
We drove leisurely. No hiking or boat trips while on our trek around North Island. Waitomo Caves for the Glow worms. View Lake Taupo and Huka Falls. Rotorua Museum. Magic of the Maori. Gannets. Aquarium. Nocturnal house. Bird Sancuarty (outside of Wellington).
The stops were made based on activities between cities. Trying to start at 8am, and get to something at 10 (when they usually open), and being were we would spend the night by 4-5pm, thus allowing time for surprises and lingering if we found a need along the way.
A bit more agressive and we could have added the Bay of Islands (we did this on a cruise of the So. Pac. at a later date).
I am sure you will enjoy your visit, regardless of your decision as to N or S Island. The decision for me was the things to do on Norht Versus South Island...as they relate to our interests. That is why I suggest you do some "North Island New Zealand Activities" and then simalar for South on a search engine to find activities that will delight you.