Thank you again, shanew147, for all this contextual information and your view on what I had found concerning this Nicholas in the Wexford Workhouse and in Polldoon Townsland.
My father looked though what my grandmother (Kate's daughter) had in her affiairs and found two interesting cards:
The first is a "Remembrance of the Mission" card recalling a mass or other religious event by the Redeptuous Fathers at Taghmon in 1905. Since Kate immigrated the following year, we believe she attended personallly. [And yes, my family is all RC, all branches including the Irish one!]
The second is a Christmas postcard with a drawing marked "Ferry Carrig, Wexford" on the front (I found online that this is near Wexford Town and is the oldest Norman castle in Ireland!). On the back is hand-written "Happy Christmas from AH to John Whyte". We think this must be addressed to brother John. Unfortunately, no date nor postmark exists as the card must have been in an envelope that was not kept. AH could be the intials perhaps of a future spouse or beloved family member. It's curious to us that it would be in Kate's affairs. Unless....
In fact, Ferrycarrig is only 2.6 km from Wexford General Hospital/Workhouse... and if indeed Nicholas was at the Workhouse in his last years, perhaps he is the sender?? Perhaps my dad is reading AH for NW, easy to confound in a scrolly cursive style? I'm checking and will keep you informed of his response.
There is a third item but enigmatic: a Christmas card from "Anty" to "Mrs Whyte". Anty = Auntie, or short for Anthony, or....?
Personally, I'm not expecting many more children to be discovered. The main reason is that Kate reported to her daughter, Margaret Knapp, that the two girls left for America and the two boys stayed back in Ireland. That combined with Mary having left before Kate suggests that sister Margaret had probably already passed (or, less likely, that she was married/established and living elsewhere) and Mary and Kate, both spinsters. No mention in family stories of Mary being married at the time of immigration. It also suggests that James and John II were the only boys. My grandmother was given the name Margaret, so I'm also thinking she was named after this older sister.
I've been unsuccessful thus far in locating Mary in South Dakota records, immigration records, Irish marriages and even in Texas, to where she reportedly moved and was never heard from again. I like your idea of looking for information on the other siblings of both Kate and her father, Nicholas. Thank you once again!