While Patrick is not in the photo, Kate is in the front row. After consulting with my mom and her cousin, we have confirmed that the back row from left to right is Ignatius (Nash with a long "a"), John Francis (Jack), Mary (Aunt Mamie), Richard (Uncle Dick), Leo (Dr. Leo Flanagan, M.D. when he was a kid.). The front row from left to right is Edward, Agnes, Kate, and Louis (Uncle Louie).
A couple of things that my mom and her cousin shared to confirm the people in the photos. First, they know what Jack looks like. We've got photos of him. My mom and her cousin personally knew Aunt Mamie (Mary), Uncle Dick, and Uncle Louie. My mom said that Louie was extremely short. This may be why he looks younger than Leo. Leo was actually two years younger than Louie. The way the photo was taken it would appear that Leo is actually further forward in the photo than the rest of his siblings in the back row.
Also, we can all identify Kate! I wonder why Patrick was not in the photo. He was still alive because Anges is there and still alive. This photo is estimated at around 1895. LOVE IT!