You can get year-long travel insurance form your travel agent. It may even be cheaper if it is only to apply to one country.
I’d be careful about doing the working holiday. The reason is that iof you intend to teach English, no employers are going to know what the hell you’re talking about. They’re going to insist that you need a work permit. If the police raid your school, they won’t have heard about this either.
What you need to do is ask the Taipei Cutlural Office (or whatever it’s called) in NZ/OZ to give you a nice letter of introduction in Chinese explaining the working-holiday scheme and making it clear that you don’t need a work permit to work.
I’d be interested in hearing how it works out for you.