On the 1880 census image of Thomas G. Martin (line 45), his name follows Dusel's (line 13-24) and I would say our Kate is "Kitty" in this group of Dusel's.
George Dusel Self M 19 California, United States
Sarah Dusel Sister F 14 California, United States
Fanny Dusel Sister F 10 California, United States
No Name Dusel Sister F 1 California, United States
Swanie Dusel Cousin M 18 California, United States
Isabelle Dusel Cousin F 8 California, United States
Dock Dusel Mother F 50 California, United States
No Name Dusel Cousin F 0 California, United States
Beecher Dusel Cousin M 4 California, United States
Kitty Dusel Sister F 17 California, United States
Dick Dusel Aunt F 45 California, United States
Minerva Dusel Cousin F 13 California, United States
GS Film Number: 1254083 , Digital Folder Number: 004239995 , Image Number: 00198