The ES Downtown / Lakefront has on site parking, either self-park or valet.
$30 or $33 a day, I can't recall exactly. We self parked and it worked out well.

Staying at the Drake I think they ask for 20-30 minutes, if I recall correctly. Of the dozen or so times we called for the car we only had to wait for about 5-10 minutes for it, just once. The doormen and other valet guys kept the kids entertained so the time passed quickly.
One of the doormen had offered to come get us if we had wanted to wait inside or up in the lobby area, it was about 30 degrees out. I thought that was nice of him. We opted to stay outside as the kids were restless and they could run on the sidewalk.