The Cass is a decent hotel with a decent price for downtown Chicago, just don't expect anything fancy. Several OTOL members stayed at this hotel for the Windy City Rail Fest in 2003. Back then a continental breakfast was included in the price. You can find a few comments about the hotel in the report located
here
That said however, consider that overnighting in Chicago will not only cost you the dough for a hotel room, it will also raise the Amtrak fare. One way from NY to LA for example costs $133 without an overnight stopover in Chicago. Adding an overnight in Chicago raises the fare to $203 a 70 buck increase.
The only other option that you'd have, if you don't want to overnight in Chicago, would be to try for an on-board upgrade. If you're traveling during a peak travel time, then you probably won't have much luck with this. However if you're traveling off peak, then you might have better luck.
Once onboard the train, ask the conductor what it might cost to upgrade to a sleeper. You can try this out of NY on the first train, but the odds are better that you'd get a sleeper on the train out of Chicago. The train out of Chicago should also be cheaper.
You can ask what it would cost for two nights, but that might be more than you'd want to pay. I'd try for just the last night of your journey. The conductor might let you upgrade for the first night out of Chicago, but he might not, since that would mean you'd need to come back to coach for the second night out of Chicago.
There is no way to accurately predict what one night would cost, but I'd guess at least $120 or higher. Also consider that getting a sleeper includes meals in the dining car, so that saves you money in the cafe car.
Finally remember that if the train sells out, then you'll have no chance of an upgrade. So if a shower and a good nights sleep is a must, then overnighting in Chicago or buying a room up front on Amtrak may be you're only choices.