birth Catherine O'Brien 198 between 1805-1815(because O'Brien is one of the most common surnames you would need something more to narrow this down)
birth Timothy Collins 2 in co Cork 1838,1843
birth Cornelius Collins 2 in co Cork 1860,1863 (in the case of both sons it may be that the first child died young and they used the name on a second child which was common)
Between 1825-1865 there are 99 children born in county Cork whose father was Jeremiah Collins and of those 22 of the children had a mother with the first name of Catherine so there are very likely more siblings.
I also looked at marriages between 1820-1840 for Jeremiah Collins and found 9; co Cork1828,1830,1835,1838,1839,1840 co Limerick1827,1832 co Kildare 1839 none of these had the spouse of Catherine O'Brien.However,if a name is spelt differently or a nickname or other forename is used it won't come up.
On familysearch.org I found this death cert: Tim B Collins died 24 Dec 1924 in Hartland,McHenry,Illinois male age 82 birthdate;4 May 1842 birthplace;Cork,Ireland father;Jerima Collins born in Irelandoccupation:farmer mother;Katheren O'Brien born in Ireland burial took place 27 Dec 1924
I could not locate a death cert for Jeremiah nor Catherine in Illinois or Cornelius in South Dakota. also,found a listing in 1865 Illinois State census; Jeremiah Collins Dunham,McHenry,Illinois there are 2 white males and 3 white females in household meaning the family arrived prior to 1865.I would check the castlegarden.org site as it contains passenger lists for New York prior to 1892 when Ellis Island opened.
I would contact the following to request copies of the death certs for Illinois;http://www.idph.state.il.us/vitalrecords/index.htm.They maintain a list of county clerks who hold vital records prior to 1916.The clerks also maintain copies of marriages and you can probably find Timothy and Cornelius there. For Cornelius' death cert contact http://doh.sd.gov/vitalrecords/order.aspxhttp://doh.sd.gov/vitalrecords/order.aspx which deals with South Dakota records.
One other thing to try is to see if perhaps Jeremiah Collins was naturalized in Illinois.I came across several Jeremiahs who were naturalized in Illinois so if there is a National Archives branch near you look at their naturalisation records to see if perhaps he was.These records can give some good info.Hope this helps,Kim