As far as I can see Archibald is listed as Lessor of all the properties listed on Griffith's Valuation, suggesting he did not live in Co. Kildare at the time of the Valuation (1853 in this area), there's doesn't seem to be any sign of him anywhere else as occupant on the valuation.
There's a Dublin probate calendar entry for 1864 Administration for an Archibald Billing which shows him dying in New Zealand at a place called Waiweraere Hotspring near Auckland as a widower in 1862. Could this be your Archibald ?
A previous topic which I presume relates to the same Archibald Billing mentions New Zealand and that his wife's name was Elizabeth, and that they were likely Church of Ireland. I dont see a likely marriage for the couple in civil marriage records (which started 1845 for non-Catholic marriages), so looks like they either married before the start of civil registration of marriages, or married outside Ireland.
I think the best place to start a search would be with the baptisms of the Irish born children mentioned in the other post - can you post their names and approx. years of birth ?
I did not see any good matches in the records currently available online, but keep an eye out for additional Church of Ireland parish records coming online, and hopefully a baptism would give you residence at the time, or at the very least a parish.