If you are going to sightsee DC or enjoy the waterfront and historic Alexandria, you won't need a car since you can use the King Street metro. Here is a link to a thread that discusses parking and staying at the Monaco
One update since it's two years old is that as far as I know, the city garage prohibits overnight parking and charges an additional $25 fee if the car is there when they close. It's confusing because it lists a fee for 24 hours but I'm going by Yelp reviews and comments. tripadvisor.com/ShowTopic-g30226-i572-k36467… If you want to visit Mt. Vernon, then it would be easier to drive there since there is no easy public transportation.
Take a look at the upper right hand side of this page to see some suggestions for a nice day . click on Day trips, urban exploration that was composed by Karen, our DE who lives a few blocks from the Monaco