I have driven from D.C to St. Louis before but I left D.C around 5 PM stopped somewhere in Ohio, grabbed a cheap hotel and then finished up next morning. This wasn't ideal but its not a bad drive, plenty of food/gas stops along the way. If you have to work the day you leave I would just be packed and ready to go once you clock out. The nap won't do you any good.
I've done many 900-2000 mile solo drives and IMO the easiest option is almost always leaving around 5 or 6 AM and simply driving with minimal food/gas stops until you hit your destination or 9 PM. Driving through the night is extremely boring, drastically increases your chances of being pulled over by cops, and brings all the creeps out at gas stations, etc. And it can be dangerous, as there seems to be a weird tendency to fall asleep right around dawn.
Pack a small cooler with sandwiches, crackers, oranges, a gallon jug of water and try to get a book on tape/ CD/ mp3 or something. Music gets boring after a while.
Cruise control at 5-7 mph over the speed limit the whole way. Be my guest if you want to try and drive faster. Speed limit is 65 in Illinois and you will get busted if you speed going across southern Illinois with out of state plates.
Is there a reason you can't fly from D.C./ Baltimore to St. Louis or K.C.? Weekend car rentals are cheap and if you're going out solo the difference between driving/ flying is usually not that much depending on ticket price vs your cars average gas mileage. If you've got a nice car, why put a bunch of miles on it and if you've got a beater why chance a breakdown in bufu Egypt. Also, the drive back with a wedding drunk well liquor and champagne hangover will suck.
Sorry to sound like a mother hen but I've done a lot of long road trips and some of them have been pretty miserable.