Emirates is only responsible to get you to your ticketed destination on the EK ticket. What happens after you get there (even if delayed) is not their responsibility. You booked way too short of a connection to begin with. You probably should have planned on staying in AKL for the night.

How long after scheduled arrival in AKL did you get there? I believe you would have had to been delayed at least 3 hours (or is it 4?) getting to AKL to get any EU compensation anyways.