Glad to hear your daughter is doing well and hope its all smooth sailing with the cast etc.
Couple of things to consider in relation to private health insurance. Its great to say you have your choice of doctor, but how many of us really know who is good and who isn't. If I was still in Victoria, where I used to work clinically I could find out, but here in SA, where I have never worked as a nurse I'd have no idea. We generally rely on the recommendation of our GP, who may have lots of reasons to recommend someone.
Your choice of private hospital is dictated to some extent by the hospitals that the doctor works at - they don't have admitting rights at every private hospital. Private hospitals don't really consider their clients to be you (the patient), they spend most of their effort attracting the doctors (who then bring in the patients).
If you are seriously ill you will quite likely need at least some care in a public hospital, only the very biggest private hospitals can do the really complex stuff, and there are some things that they just don't do. If you are elderly with complex medical problems you will probably be in a public hospital, private hospitals try to minimise looking after complex medical patients, its too hard to make money on them (and even the not for profit private hospitals still have to make money - they should be called 'not for distribution to shareholders' because they still have to make money).
If you are in smaller private hospital and have a medical emergency (like a cardiac arrest) they will call an ambulance and ship you off to the nearest public A&E. I used to be a unit manager in small private hospital and that is exactly what we had to do on several occasions.
Very few private hospitals have kids wards.
Don't get me wrong, you get very good care in private hospitals, the nurses, doctors and other health staff all came through the same universities. But there are differences in the two systems that mean that they are not always interchangeable.
Basically the funding model for private and public care is totally broken, the ones doing the best are the private health funds (private health insurance industry made about $1.6 billion profit last year).