Lawn Newbie needs help

Posting from New Zealand so our springtime ends at start december and autumn kicks in around Feb. These are our growing seasons ( apparently ) as I moved from the UK last year. /
Just moved into new house a while ago and eventually got around to addressing the lawn areas. I. After cutting back the lawn I can now clearly see about 40 patches of dead grass where the previous owner let their dog ( a bitch ) pee everywhere. Around these dead patches the grass appears to have gone mad and I have posted a couple of pics. It's very elongated, almost wiry and grows sideways over the holes left in the grass..
When I dethatched today, it brought up stalks from 3 to 6 inches long all over the place!! I am talking around 5 metre square areas covered in the stuff. It appeared large areas of the grass were made up of this long stemmed grass, not the nice stuff I initially thought.
The majority of the grass generally appears rather "rough" in texture. I suspect as with most newer properties in NZ it is readylawn put down around 4 years ago. I have also enclosed a picture of a couple of small areas.
Given the scope of the problem I was keen to hear your thoughts and advice on wether I should kill off the grass towards the end of summer and re-seed or will I get good results working with what I have.
Thanks for any advice. Please ask anything you need from the photos.

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