We can’t of course, NZ has no mechanism to “impeach” / remove a PM.
In effect NZ is an elected dictatorship, without the checks and balances built in to (for example) the US system. The PM (as long as he retains support in Parliament) can get away with anything. It’s basically an honour system. We expect, and have so far (mostly) received, reasonable and responsible behaviour. Acts are supposed to have consequences. When politicians are caught our screwing up or lying they are supposed to resign. There is supposed to be a measure of shame.
Those old rules no longer apply. The honour system is broken, exposing the flaws of our political model. Key is a rogue PM, who will say and do anything for power. He can only be removed by his own supporters in Parliament, and (of course) that is never going to happen, because they derive all their power and perks through him. Every single one of them will keep their heads down and keep banking the cheques. So we are stuck with an amoral liar for the next 3 years.
I think the US model is badly, paralysingly broken. But now our honour system is broken too. We need something in between. We the people need to reform our political model so that there is a realistic constraint on the elected dictatorship.