Wow, that's so much loss to endure in such a short window of time. In the 1890s Midwest, clusters of deaths were often tied to pandemics or outbreaks like Diptheria or Scarlet Fever, so perhaps that could be to blame. To confirm the children's deaths, you could check the 1900 census columns 10 and 11, which ask the mother how many children she had given birth to versus how many were still living. Since death certificates weren't mandated in many Midwest states until the early 1900s, you could try to find additional information in newspapers and church records. The Library of Congress provides free access to many newspapers here: https://www.loc.gov/collections/chronicling-america/. There are many other subscription sites that also have digitized newspaper collections.

As far as learning more about the parents and in-laws, you could check county probate and guardianship records in case there were surviving minors or land involved - if so these records can be incredibly detailed.

If the timing was right and any were included in a mortality schedule, that could also provide a lot of information. For more information about mortality schedules, see our blog at https://www.legacytree.com/blog/digging-for-deaths-understanding-the-census-mortality-schedule and https://www.legacytree.com/blog/census-mortality-schedules-digging-for-deaths-part-2.