The "no. of days" depends on your activities here. Not sure if you plan to only venture in & around Auckland - if yes then 2 weeks would be good enough.
In case you plan to recce for job then add another week or so for meeting agencies/interviews etc.
Serviced apartments in Auckland CBD have a small kitchenette and usually short walk from public transport. These range from $300 to $400 p/w and are offered for minimum term of 4 weeks.
Have a look at AirBnB website if you can find more options for short term rentals. There are number of backpackers accommodations within CBD too.
Hope this helps for a start