I used to bike the lakeshore from Uptown to Millennium Park on a regular basis. It may be a bit longer, but it's blessedly flat, and it's gorgeous. Also, at Millennium Park there is (used to be?) a bike station where you could rent a locker and shower and lock up your bike securely/safely/out of the weather. It was two blocks from work; I loved it.
One problem you'll run into on the Lakeshore multi-use path, though, is headwinds. I used to check the wind every day before I left for work or left from work; above a certain windspeed, in the wrong direction, and I'd just take the bus. It saved me a lot of rolling backwards.
That said, I've seen people biking along Michigan Ave with no problems in almost every kind of weather. I think it's doable if you're not traffic-shy.