SPRINGFIELD — Justice Anne M. Burke was selected to become the next chief justice of the Illinois Supreme Court.The Chicago Democrat will step into her role on Oct. 26. She will be the third woman to serve in the position.Burke was first appointed to the high court in 2006, then won election in 2008 as the unopposed Democratic candidate. Cook County voters retained her for a second term last November.Before joining the Supreme Court, Burke sat on the 1st District Appellate Court bench for 11 years. She’ll be …