My husband and I also married in Australia and then we registered our marriage to Suku Dinas Catatan Sipil in Grogol, Jakarta when we came back to Jakarta.
Basically, your wife has to registered the marriage within 30 days once she comes back to Jakarta. You don't have to submit the real Certificate from the Births, Deaths & Marriages office but you can use a copy of the certificate the marriage celebrant gave you on your wedding day.
There are lots of documentation that needs to be submitted such as a copy of the certificate from your wedding celebrant, your birth certificate and passport (as far as I can remember it needs to be translated), your wife copy of KTP, a copy of KK, a copy of her birth certificate and passport. For further information, I think it would be best if your wife or someone from the family comes to the Suku Dinas Catatan Sipil in Grogol for the details. There's no need to pay any fees.
Hope it helps.