What is the ruling on traveling to Australia in Ramadan to recover jewelry and finalize important documents, knowing that it will cost a large sum, and istikharah was performed without a decision being settled upon?

There is no objection to traveling to the city of the Messenger of Allah, peace and blessings be upon him, but it must be with a mahram (definition). A mature son can serve as a mahram. The fact that the travel is in Ramadan (definition) does not necessitate unlawfulness, but it is better to choose another time for fasting in the city. It is permissible for a woman to pray in the mosque with lawful conditions, though her prayer in her home is better. Istikhara (seeking guidance from Allah) in permissible matters is a good thing, and its result is known by the facilitation or non-facilitation of the matter.