Check in church records, census, naturalization petition or social security index. Anyplace that they might put a name of a town. Without that name, you can not go far. There is a possibility that they came from Galicia region which you might need to post on Poland and/or Ukraine boards.
Austria-Hungary it was at the time your ancestors came to America. After the fall of this empire several new countries were formed and one was Czechoslovakia which in 1993 separated into Czech Republic and Slovak Republic.
Do you know what language they spoke or church attended?
If they do not show up on Ellis Island website than they might have passed thru Baltimore, Philadelphia, Boston or Galveston.
Check out the spelling LENYKO, LANYKO and LONYKO on the Ellis Island. They all could be Lenko.