Cairns will be hot and muggy...but the Whitsundays not as much.
For luxury try:
Emirates One & Only resort at Wolgan Valley - some good activities for the kids. Only 3 hours from Sydney and for your limited time will probably be the best bang for you.

If you can escape Sydney and want a unique experience try Lord Howe Island - Capella Lodge is the go to there.

A week in Australia will either get to a heap of fly there, flew out or 1 maybe 2 great places to explore.