If the OP wants to use the miles to get an award ticket on Skyteam, yeah KE is the only choice you have to AKL all the way.

If the OP is paying for a ticket, UA+NZ (Star Alliance) and EK are the best options in my opinion. Or AA+QF (oneworld).

CMH-somewhere-SFO/LAX with UA then NZ to AKL.

CMH-somewhere-NYC-DXB-SYD-AKL with EK. I think EK has good fares ex-USA to Australia and NZ via Dubai, albeit a long way round!