I did not find Ernestine in MOMS (Minnesota Marriage Index), or on Find-a-Grave, or in Minnesota Death Index. I found children in Minnesota born in right era who had a mother with maiden name of Thiele (first names of mother are not given, Minnesota Historical Society Birth Index). I then searched Ancestry.com census database to try to match those children with a mother named Ernestine. Did not find her. I did not do the same for South Dakota.
Some other volunteer researcher many find her. If not, the next step is to post all the information you can. I’ve made a start, below.
Name of Leske in this family may also be spelled Lieske, is that right?
From Ancestry.com: 1880 CENSUS Waconia, Carver, Minnesota Arnold Thiele 35, born about 1845 in Prussia [1842?] Ernestine Thiele 34 born about 1846 in Prussia Carl Thiele 16 born MN Johanna Thiele 13 MN August Thiele 12 MN Herrman Thiele 11 MN Johanna Leske 56 mother-in-law, widowed, born about 1824 Prussia Ernestine Thiele 3 born about 1877 in Minnesota Aaron Thiele 1 MN
The family seems to have moved to Roberts County, South Dakota, is that right? Do you have census transcriptions you can post? Other information on the family you can post? It is not uncommon for siblings to found near one another (or even to live with one another), so the location of a sibling could be a clue. Or perhaps you've exhausted those possible leads....?
Ancestry.com gives “Ernestine Block” in Wisconsin as a hint/suggested match for the Ernestine born 1846, but I did not follow that up.
Okay, that’s all the time I gave this…to be continued!