To Amtrak's credit, the rebooking line when we finally arrived in Chicago was long, but moved quickly. As I waited, a representative went down the line with new tickets, and hotel and food vouchers for everyone who had missed their connections. I waited maybe 15 minutes in total.
Another important note: Amtrak did offer hotel vouchers for those of us who needed to wait overnight. I had rescheduled meetings for early the next day as a result of the delayed train, and didn't feel like schlepping to The Loop during rush hour from Midway Airport. It also proved a worthwhile moment to put travel insurance to the test.