Julie-Anne,
Unless the records for Sligo Co Infirmary have survived, and you can get access to them (they won’t be on-line), it may not be possible to say for certain whether it’s the correct lady. No townland is given on the death certificate and the informant wasn’t family. But if his wife died, it wouldn’t be surprising if Robert then left his infant daughter with his sister (a typical sort of arrangement in those circumstances) and headed off. That could account for her later bringing Annie to Australia.
Perhaps worth checking migration records on Findmypast or Ancestry to see if you can find Jane James and Annie travelling to the US (1878 to 1900).
If Robert and his sister Jane were born in Riverstown, it’s in the parish of Taunagh. Their Church of Ireland records start in 1830 and are in the RCB library in Dublin. They are not on-line yet, so you would need to get a Dublin based researcher to access them. But you might find their baptisms and that of Robert’s siblings, that you mention. The records for this parish are highlighted in green in the Church of Ireland guide to their records. Green means: LOST - Parishes the records of which were lost in 1922, but for which subsequent materials were located and lodged either in the RCB Library or PRONI. Normally this colour code is followed by an RCB library or PRONI entry.
So it looks as though the original set of records was destroyed in the fire but that a duplicate set existed somewhere else and is now in Dublin. There is no RCB library reference shown in the records.
I found what looks to be Robert’s father’s death in May 1898, aged 91. (I mentioned he too was a police officer, as per the 1871 marriage). This looks to be his death in Collooney. So that links in with the information that the family did live there.
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1898/05831/4651070.pdf
His wife was still alive and the informant was his niece Jane Breach of Belfast. John Jones wife Hanna(h) died the following month, again at Collooney. She was only 66 so might be a second wife. Informant was again Jane Breach, this time of Collooney. So that clue took me to John James marriage to Hannah Tighe in 1880. Both had been married before. His address was Riverstown.
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1880/11041/8038443.pdf
Reflecting on that, I wonder if the death of Robert & Jane’s father (and their stepmother) in 1898 was the trigger for Jane James to leave. Had she been at home looking after them? After they died, she was released from that obligation. Did she head to the US and then Australia? Did she go to the US because she knew Robert was there? Presumably father and daughter might want to meet?
Finally, Hannah James left a will. The executor was Jane Simpson of Ballisodare, the sister.
http://www.willcalendars.nationalarchives.ie/search/cwa/home.jsp
Jane Breach living in Ballysodare in 1901 with Hannah’s sister:
http://www.census.nationalarchives.ie/pages/1901/Sligo/Collooney/Ballysadare/1682649/