For pure miles but staying in F where possible I guess off the top of my head it would be something like:

SYD-HNL-ORD-ORF-ORD-LHR-DXB-LHR-IST-LHR-HKG-NRT-HKG-NRT-HKG-MEL-PER-MEL-DRW-SYD

You're a sector short of the maximum 20 because you only have 3in NA.

If you'd be prepared to go J for HKG-AKL then you could possibly get a few more miles that way. You do know that SYD-HNL doesn't have F either?

AA F on ORD-LHR is fine if you get one of the Flagship Suites - in my view these are the best F seat on OneWorld. On average the BA service might be slightly better, but there's not much in it. In fact with kids AA might be better because all the cabin crew will probably have grandchildren of their own