New Zealand North Island in 5 days, including Wellington, Rotorua, and Auckland?

My family of 3, including myself, are visiting the north island of New Zealand in the beginning of July. We'll be flying from Australia to Wellington, staying 2 nights in Wellington, 1 night in Rotorua, and 2 nights in Auckland. We would like to explore the north island with a car hire as well, driving from Wellington to Rotorua (and possibly visiting Tongariro National Park and the Waitomo Caves (glow worm caves) along the way), staying a night in Rotorua, then drive up to Auckland. Would this be a do-able itinerary? What are some suggestions for how long we should stay at and what to see in Tongariro National Park, the Waitomo Caves, and Rotorua for our itinerary? I'm open to other suggestions as well as far as what to see and itinerary as well. Thank you!