How long have you had already on your WHV? I see you're from England, so I'm asking if you've had the full 23 months you could have put in for.
http://www.immigration.govt.nz/migra...idayscheme.htm
Yes, there are visas which allow you to stay and look for work after graduating from an approved course of study.
http://www.immigration.govt.nz/migra...tudytowork.htm
There have been several forum members who have mentioned the post-study work visa.
https://www.google.co.uk/#q=site:+en...te:www.enz.org