First of all it doesn't rain nearly as often as it does on ER. Late December usually begins our really cold and snowy days. That being said, last year I believe the last week of Dec was in the 40's. But, downtown is always very cold due to the wind whipping in off the lake. You;ll need a good winter coat, hat and good gloves at the least. A scarf to wrap around your face is a good idea as well. Snow? It's hit or miss that time of year. Either way play it safe and bring a comfortable pair of warm and slush proof shoes/boots. Coming between Xmas and New Years gives you alot of options. Go see the animals and the lights at Lincoln Park Zoo. Take a hansom ride down Michigan Ave in the evening and enjoy the lights and sights (very romantic). Millinium Park will have ice skating - rent your skates there. Lots of great theatre in the city - See if you can get tickets to Wicked.
Quite honestly, Gurnee Mills never impressed the heck out of me. Personally I think it's a run of the mill (no pun intended) discount mall that is spread WAAAAY to far apart. Deals? Nothing that has ever impressed me. From downtown it's about 1 hour. Unless your really into hunting for "bargains" I'd just stay on Michigan and on State Street and window shop. Why rent a car? Take the "L" and explore our Indian neighborhood on Devon, Chinatown on Cermack (also known as 22nd). Are you here on Sunday? Make reservations for a Gospel Brunch at the House of Blues in Marina Towers. Or have lunch in the Walnut Room at Marshall Fields under the big Xmas Tree (a Chicago tradition). I believe the Drake Hotel still does an afternoon tea. I won't even go into the museums, Navy Pier, etc etc.