Hello Angela,
I am researching the Lada family for some friends. I don't have any information on your Uncle Joe in Chicago but I think I have some information which will help you trace your roots.
Stanislaus "Steve" Lada fell and struck his head, dying in the ambulance on September 1, 1923. A pdf of the death certificate is on-line at the Missouri Secretary of State website. The death certificate informant looks like Anthony Lada although I might have read that wrong. Stanislaus or "Steve's" father's name is listed as Joseph Lada from "Russia Poland." The mother's name isn't given. Terri Hoelting, in her posts on this message board, suggests a different father; if she she is correct, the informant for the death certificate gave incorrect information which sometimes happens.
Julianna Lada Struminski is listed in the 1920 Federal Census living next door to Adam Lada. In 1941, after the death of her first husband, Julianna told/showed documents to the priest officiating at her second wedding at old St. Casimir that she was the daughter of Joseph Lada and Marianna Jablonowska (my reading of the name is uncertain) and was baptized in Mostkowo, Pow. Ostroleska, Grenia Lomza (my reading of the names mentioned might not be correct). This seems to be the same as present day Mostkowo, Gmina Lukta, Pow. Postroda, Warmian-Masurian Vovodeship in Northern Poland.
If the information on Stanisaus/Steve Lada's death record is correct, his father Joseph Lada might be the same Joseph Lada which Julianna Lada Struminski Przstup mentioned as her father at the time of the second marriage. You may also have the name of a town in Poland where you can search for your ancestors.
Happy hunting,
Jim Foster