Will that be a day-trip from Auckland? It's a long drive but do'able. I suggest SH1 to Taupo, then SH5 over to Napier. Plan short stops at Hamilton lake, then a longer break at Tirau &/or Taupo, followed by Napier.
Let you children determine how far they can tolerate the car journey. Auckland to Hamilton lake is easy, so too is the drive through to Tirau. It's a big push to Taupo, being a 4 hour road trip by then. Hopefully the last bit will progress well. Huka Falls is a great attraction and a chance for some fresh air and exercise.
If you are looking for a nice walk, check out Blue Springs (Te Waihou Walkway). If you park at the Leslie Rd carpark, the springs are only a 20-minute walk. This is located south of Matamata and north of Rotorua.
As one can cannot usually "get into" a motel before
2 pm the "mid-way" concept mentioned in replies 4 & 5
may "not make sense".