Gingee,
I believe there is a parking lot near Lou Mitchell's restaurant--I'm sorry I don't have specific directions, but if you cross the street from the Canal entrance of Union station and walk straight to Lou Mitchell's, you should see it. I believe parking is between $10 and $20 per day.
With that said, I agree that leaving your car in the downtown area long-term could be a bad idea.
I don't drive, but I think it could be potentially cheaper for you to park in the suburbs. I like am partial to the Naperville area and have stayed there numerous times but never had any experience with parking.
You might want to click on:
http://metrarail.com/metra/en/home/maps_sc...sf/station.html
and call Naperville, Lisle, Downers Grove, La Grange stations and ask about any long term parking in the area, particularly secure lots.
Click on each station for the phone #.
You could drop your car off and probably take a short cab ride to the metra station and ride the train to Union Station.
I was also curious so I called a couple 3 and 3.5 star hotels in the area and asked them if they would allow parking in their lot as long as you were a guest at some point. They said it should be OK.
For 10 days, I would probably reserve 2-3 nights at the hotel, several days apart. However, I recently reserved a couple nights at the 3.5 star hotel and paid around $45 per night tax included, so it would still probably fall within your budget. Plus, if you book two of those nights at the beginning and end of your journey, you can actually use the hotel and take the shuttle to and from the metra station. You can PM me if you need more info.
Using the hotel lots is only one option. It's possible you could find long term parking in one of those areas for cheaper.