What would be the income tax rate be if it were 'flattened'?
Having worked on these issues myself for quite some time, I am not aware of any study on this for New Zealand. For a back-of-the-envelope calculation of the revenue-neutral flat tax rate, you'd need the following:


The tax base $Y$, i.e. the amount of income subject to income tax. Most probably, the New Zealand Treasury has published statistics on this. Note that total taxable income is different to total earnings which are reported in national accounts.
The number of taxpayers $N$.
You might want to maintain the current exemption $E$ which remains untaxed for each individual.


With current income tax revenue $R$, the revenue neutral flat tax rate $\tau$ can be obtained from:

$ R = \tau * (Y-N*E)$

Note that is the minimal thing to do which abstracts, among other things, from behavioral effects of such a reform.