It's wonderful to hear that you have had so much luck tracing your paternal side of the family!
Before 1864 records were kept in the form of church and parish records; while some of these have regrettably been lost over the years, all the surviving records have been sorted into county-based genealogical centres
. If William Kennedy was for the County Kerry area then I would suggest trying to contact the Irish Genealogy
website who hold church records for the Roman Catholic Diocese of Kerry and some Church of Ireland parishes in Kerry.
You may also wish to contact Ireland XO
who have a lot of useful information available to help with searches.
I hope this gives you some more inspiration, and I'm hopeful that your fellow members and some industry partners will be able to help further.
If you need any more help, just let me know!