Hi Dicky,
I'm afraid I can find no record of a Jane Byrne born in Co Kerry in 1856. Have you got any more information about the possible part of Co. Kerry from which she originated?
I did find a record of a Thomas Byrne who farmed land in Inch West, Galey - near Ballybunion, around 1850: http://www.askaboutireland.ie/griffith-valuation/index.xml?action=doNam….
While I was unable to find a record of him having children, I did find another family - Michael Byrne and Bridget O Brien, in that townland who were having children at this time. One of these was named Thomas, born 13/01/1861. Members of this family were still living there at the time of the 1911 Census:http://www.census.nationalarchives.ie/reels/nai002492996/
These were the only possible relatives of Jane that I could locate.
Hope this helps.
Joan