Some of the major magnet hospitals in Chicago are Rush, NMH, University of Chicago, Children's Memorial, RIC, and Advocate Illinois Masonic. There are a few more though. The hospitals in Chicago are looking for experience, certifications, BSN, leadership, and community service. Make sure to highlight these areas in your resume.
Transferring your license is a pain, but easy enough. You will need to go to this website:
Click on the appropriate "written application" link for your licensure. Then, choose the "endorsement" application. Print and follow the steps for the endorsement requirements. Make sure to send everything in a neat package, otherwise receiving your license may be delayed.
Hope that helps