This situation has actually occurred in my former husband's family while he and I were married. My husband grew up as 1 of the youngest of 7 boys. His father had a relationship with a lady before he married my mother-in-law which resulted in a daughter he never knew about. The adult daughter's family (a sister, I think) contacted 1 of my older brothers-in-law about her which, of course, was passed along to his parents and the rest of the family. Dad didn't want to talk about it but Mom was very open to meeting her. I think Mom's attitude about it helped Dad to overcome his misgivings about the situation. The family embraced her and her children. They became part of the family through several visits and extended stays over the years plus calls and letters in between.
I think maybe finding a "middleman" to approach with the information, first, would be a good idea. I would try to find someone who knows the family fairly well that you could talk to about the situation and get some idea of whether approaching the family (or a particular person in the family) would be okay or how the family might react to the news, etc.