This is probably not relevant info, but I couldn't help mentioning it, because the names Henry King and John King are somewhat common in one branch of my family which lived not too far away from there, and there's one Henry King who would be about the right age to be your Henry King, although his father's name was James. His grandfather was named John King, however. The other complications are that they were Catholic, and lived about 6-7 parishes away in Kilcolman parish in County Mayo. Still, living that near, they may well be related, and some family members might have converted (as the Ryan family obviously must have done at some point, since Ryan would normally be a Catholic name).
I just went back to check the name of Henry's father in the parish register, and it does appear to be James, but it's one of the annoying abbreviations which the priests used back then. They often used "Jas" for James and "Jno" for John, but in this case it's just "Js" (unless there's an "a" squeezed in there which I can't see). I rechecked the baptismal records for his brother, Nicholas, too, and there it's clearly "Jas".
"My" Henry King was baptized in Kilcoman parish (Claremorris) 25 September 1841, and we have no records thus far as to whom he married. if you want to see more about that branch of the family, you can see Henry in the (free) Family Search database under his identifier # LTDP-CM8. Here's a link to Family Search, in case you're not familiar with it: https://www.familysearch.org
There are a lot of our King records in that database, because I've made contact with King relatives living in the UK, and we've combined our info there. I'm descended from "my" Henry King's great-uncle, Nicholas King (Nicholas is another name running in the family), who had also named his eldest son Henry. That Henry (born 1837) would in fact also be of an age to match your Henry, but we know that he married a Mary Durcan in Kilcolman parish, and they were once again Catholic.
In case you've done DNA testing, I and several second-third cousins in my King line have also been tested, and we'd be happy to compare results.