I wouldn't recommend it. (NB - to get 5500 per month at 15 per hour, you'd need to be working a 90 hour week!)
ChCh is very expensive currently.
Say your maths are right, and you have 4940 per month. Would be tight.
Accomodation for somewhere acceptable will be about 1940 of that, leaving 3K. Food - at least 900 a month if you only have one child and go to Pak N Save etc.
Power - 250 a month, Phone, say, 150 a month (mobiles, landline)
Car - fuel to get around, say, 250 a month if you just go around chch.
Clothes/Kids Stuff/Holidays - say 300 a month on average
WOF/REGO/Insurance etc - say 150 a month on average
That in theory leaves you $1000. But I've kept everything low (no iphones, no broadband, no sky tv, presumes you have capital to buy a car outright, presumes you have capital to furnish a house outright, presumes only one car, wife does nothing in the day, no entertainment, no savings etc. etc.).
In reality, your costs will be higher, and you'll have very little disposable income.
But on the other hand, life is an adventure, but I would point out that although I love ChCh, there's a lot here missing from the UK, and living here and existing - i.e. just getting enough to cover costs - would be less satisfying than in the UK, e.g. you have no live sport if you like football, pubs v expensive etc. etc. - more be a move if you like the outdoors, skiing, swimming, beach, tramping etc. (all the stuff for free in NZ)...