I haven't found a marriage or death certificate for Ernest Housch and wife or birth/marriage/death certificates for the children. Those documents should have the parents names (including Kate's maiden name, we hope), and could be no doubt be ordered from Harlan County.
1900 Antelope,? Turkey Creek &? Scandinavia Townships Ragan village,? Harlan,? Nebraska Karl W Hoesch47 Katie Hoesch46 Emilie Hoesch21 Mary Hoesch19 Earnest Hoesch17 born Oct 1882 in Germany Henry C Hoesch12 Paul Hoesch8 Bertha Hoesch6 Anna Hoesch4 Albert W Hoesch1
1910 Turkey Creek, Harlan, Nebraska Ernest Hoesch27 Katie Hoesch21
1920 Turkey Creek, Harlan, Nebraska Ernest Hoesch 37 immigrated 1883 M Katherine Hoesch 31 immigrated 1892 Esther A Hoesch 9 Walter R Hoesch 0[3/12]
1930 Turkey Creek, Harlan, Nebraska Ernest Hoesch47 Katie Hoesch41 Esther Hoesch20 Walter Hoesch10 Lela Hoesch8
1940 Turkey Creek, Harlan, Nebraska Ernest Hoesch57 Katie Hoesch51 Walter Hoesch20 Lila Hoesch19
Looks like Lila/Lela got recorded in two places, Turkey Creek and Salt Lake City
1940 Salt Lake City, Salt Lake, Utah, USA Ralph Hardy 27 head of household born in Nebraska Esther Hardy 29 wife bornin Nebraska Barbara Jean Hardy 2 daughter born in Nebraska Lela Hoesch 19 sister-in-law born in Nebraska