I'm in Qld, but here's a little info and will leave the rest to someone who knows the are better.
Here's the link to school dates for this year in NSW:https://online.det.nsw.edu.au/calendar/?do=setViewperiod&categoryid=21&viewType=yearView
The school year there goes from January to December, and you'll probably have to contact the department on education to find out which grade they'd enrol your daughter in. IME, that can depend on both her age, and her academic ability. For example, she might be of an age to be enrolled in Year 2 here, but they might enrol her in year 1 if you wanted her to be, or her academic background suggested that she'd be better there, and for so short a time, or vice versa.
The enrolment application suggests that temporary visa holders can be enrolled, but you'd probably have to contact them to find out details.
Finding childcare at an institution that isn't a school would be quite difficult for a child of school age. I'm sure a Nanny would be an option - Newcastle is a large metropolitan area, and I'm sure you could find someone to care for/homeschool DD, even for that long. I would suggest that a state (public) school would be an easier option.
Be aware that most state schools have a uniform for students to wear, but you'd probable be able to get second hand fairly inexpensively to wear for that length of time. Or they might allow a more generic, but similar coloured shirt/shorts combination for a couple of weeks.
I'd probably start investigating daycare options ASAP for DS - depending on all sorts of factors, daycare options can be difficult to find in Australia.
Hope this helps!