The Apgars died in the 1890s. Elias died in 1894 and Anna died in 1895.Both of these certificates were obtained from the Illinois, Cook County Deaths, 1878-1922 on the FamilySearch site.
There is no indication on the certificates who the spouse or widow was. However, given the marriage certificate and that Elias is married and Anna is a widow and the undertaker and cemetery are the same, it seems pretty clear the certificates are for the same people.
Annie died in 1895 as shown on her death certificate.
The problem is that there is nothing in these records to indicate whether or not there were any adopted children.
The letter indicates that the Apgars adopted a Willie and a Frank, but does not provide any more details about these children. The assumption is that they were brothers, but the letter does not specifically make that statement. William Frame did have a younger brother Frank. One problem is that I'm certain why they would have been adopted. However the adoption would explain the use of the last name Apgar.
I'm working on the "next steps" in this research. That includes searching the 1888, 1890, and 1892 Chicago Voter Registrations at Ancestry.com and the 1880 census for Elias. Then I'll compare the locations I have for Elias with those for Thomas and Elizabeth Frame, William's parents.