Having condensed our 1st visit to NZ into 5 days i would stick with 1 island and remain on the North Island.
Day 1 AKL - Hobbiton.
Pre book a tour of Hobitton before you leave your home & take the tour from Matemata. Great visitor centre and free parking.
PM: Waitomo Caves
Overnight Rotorua
Day 2 Book a flight seeing trip to either white Island or a helicopter tour over Tarawerra that lands on each site and gives you time to explore the volcanoes
Evening, soak in the hot springs such as Hells Gate or the Polynesian pools
Overnight in Rotorua
Day 3 Drive to the Coromandel peninsula and explore cathedral cove and Hot water beach
Overnight in Thames
Day 5 Thames - Auckland via Miranda hot springs