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I believe that the Royal Barracks is what is now called Collins Barracks which is part of the National Museum of Ireland (which includes Collins Barracks, the Natural History Museum, Dubin and the Museum of County Life in Castlebar, Co. Mayo).
Records of those who were stationed there would have been records of the British government so the website of the National Archives in Kew, London will be of help to you where you can 'search for a person' here: http://www.nationalarchives.gov.uk/records/looking-for-person/
In relation to Leixlip, you could have a look on Find My Past (a subscription site) to see if there are any Trade Directories that name a doctor/surgeon in Leixlip but I am not ware of any records that would tell you about any work Ann did there. You could try the Irish Newspaper Archives for a mention of her. Some of my ancestors are mentioned in local newspapers as they did well in exams/got were thought to be good jobs - it is a long shot though.
The church there was built in 1833 so I would think that it was the only RC church building at that time.
She may well have been named Anastasia and called Ann all her life. I have often heard of people named 'Baby' who didn't know their actual baptism names and there are a variety of nicknames that families use that can lead to similar confusion!