Family search (famiysearch.org) quickly lists 2 entries that seem to be related to you.
Under the "Ireland, Births and Baptisms, 1620-1881 records we find....
Bridget Burke; b. 01 May 1870 to Edmund Burke and Mary MacDonagh on Inishbofin, County Mayo, Ireland (Off the top of my head Inishbofin was transferred to County Galway circa 1880 to 1885.) The birth date fits closely with the baptismal date you gave.
There is also a Mary Burke; b. 2 Mar 1866 to Edmond Burke and Mary McDonogh in Roundstone, County Galway, Ireland.
Checking a map of Ireland will show that these 2 locations are fairly close and in the same parish so it may be possible for these to be the same parents (Edmund and Mary). This could be an earlier daughter that died before the one you listed, or maybe not. Something to explore. Hopefully this link will get you to this data.
Looking up Joseph O'Kane I came across the "tree" you had started. Other then that there are a lot of Joseph O'Kanes in londonderry.
I'll see if we can wake Tommy Burke up and get anything more from him.