Where are you based in the lower North Island, and what sort of time-frame are you working in?
If you're interested in going further north into the Ruahines (north of Palmerston North), many Ruahine road-ends start quite high up, which makes it easier to get into the tussock. The down-side is the road-ends tend to lead further in from the State Highways on either side, so it could take longer to get to than many of the Tararua ones. The Ruahines also aren't too popular in winter, I think because the weather cuts down many of the options for getting far into the range, though there are still many options on the edges.
Heritage Lodge is a very accessible back-country hut that's operated by the New Zealand Deerstalker's Association and sits about 100 metres above the Oroua River. http://www.nztopomaps.com/-39.957320,176.006899,14,Map,100 It's an easy walk in and out from Peterson's Road (think 30 minutes, easy down-hill/flattish grade.
Sunrise Hut is a popular hut on the eastern side for family groups. http://www.nztopomaps.com/-39.793963,176.172209,14,Map,100 The track to the hut is very switched and well graded, and definitely marketed towards people who've not done much tramping (lots of signs about things to look for in the bush, etc), and from the hut you'll get nice above-the-bushline views with an option to walk around up to Armstrong Saddle if the weather's nice.
Longview Hut's a good hut on the eastern side, from the end of Kashmir Road, above the bush-line. It can get extremely windy up there if the weather's like that, though. (It's one of these huts with lines tying it down, and it's very noisy inside when the wind's up.) http://www.nztopomaps.com/-39.953701,176.140451,14,Map,100 Awatere Hut, also on that map, is a little bright orange hunter's hut that'd be easy to get to. It's only 3 bunks so definitely be prepared to camp, but there'd be good camping nearby. You'll need to cross the river outside the front of the hut, so if the weather's really bad take that into account.
Rangiwahia's a great hut just above the bush-line on the western side, where people used to carry up their skis, and there's a good opportunity for walking around in the tussock, and views over to Ruapehu. http://www.nztopomaps.com/-39.896765,176.001749,14,Map,100 It used to be extremely easy to get to until a big slip destroyed part of the track about 10 years ago, so a 45 min to 1 hour easy sidle up-hill is now sign-posted at 3 hours because the track detours very steeply up over the top of the slip. I've seen young children up there, but you should definitely be prepared to be carrying your daughter around the slip... maybe someone here who's a parent could chip in with how realistic this is.
Enjoy.