I'm not sure why Tommy hasn't responded to you yet but I have some questions. Perhaps they are the same that Tommy has or perhaps he may have the answers already.
Also... All that I note here is not to say that you are not related, only that we don't have all the information yet. There are many branches where we think there should be more data but ....
Bridget McDonough Burke, b. abt 1870..
Closest I could get was Mary Bridget McDonagh, b. 28 Jan 1873, but we lack further information on her. No Marriage, no death.
The one Bridget McDonagh I have in my database was b. 3 Jan 1872, married Michael Scuffle 7 Jun 1895.
I have no Bridget McDonough's.
The Bridget Burke's I have are all married in Ireland.
I have 2 siblings named O'Kane; a Mary Jane, b. 1843, and her brother Percy, b. 1850.
So the question back to you is what more information can you provide? Siblings to your grandmother? Something about her parents? Cousins?
I've tried looking at this several different ways;
1) the McDonough's that might have married a "Burke", either Bridget or her mother
2) the different spellings of McDonough/Mcdonagh
Of course the one thing I don't know is if Bridget was her baptismal name or one conferred on her as a "nick" name by her parents; always the bane of researchers.