Hello Thomas, unfortunately John McAfee, for all his kindness, probably searched the wrong province. Despite the popular miscinception that all Kanes are from Ulster, we Galway/Roscommon Kanes (more recently changed to Keane in these parts)have our own separate history here in the province of Connaught, in the Irish Republic.
I am a DNA match with a Kane from Com Roscommon, plus other Galway Cains. I have Connor ancestry also. Too bad you don't have any male Kane relations who could provide a DNA sample, because I am sure that would make your family the Connaught Kanes, not the Ulster Kanes. There are plenty of both in the Kane DNA project at www.familytreedna.com