Here are my thoughts:
1. Heading east, the LSL leaves Chicago at 9:30 pm, and there is no lounge car (just a small cafe). There is nothing to do but go to sleep. Heading east the Capitol Ltd leaves Chicago at 6:40 pm with a full dinner service in a real dining car beginning at 7 pm. After that, you can go to the Observation lounge car until midnight. This is more enjoyable than the LSL imho.
2. Day 2 of the LSL journey is all New York State, with a scheduled arrival in NYC at 6:30. This may be ok for some, but in my experience the all day ride (with an hour+ stop-over in Albany) can be tiresome. Then you arrive in Penn Station during rush hour. By contrast, the Cap Ltd arrives in D.C. around 1 pm, after serving brunch until 11. Much better.
3. If you continue to New York the same day, and you're worried about missing your connection in DC because the Cap Ltd might be late, don't worry. If you book both trains together as a single trip Chi-NYP, and the Cap Ltd is delayed, Amtrak with automatically rebook you on a later Regional, based on your actual (not scheduled) time of arrival in DC.
If you stay over in D.C., of course you don't have to worry at all about the connection.
4. Finally, you might want to consider the Acela for the DC-NY leg. It's a different experience entirely and quite nice.