What about parking at an airport like National and taking the Metro to downtown?
Just to provide a few additional data points, my approach is from the south (driving up from NC).
What about parking at a place like National airport in DC, then taking the Metro into downtown? Don't know how much parking is at National, but it's only $6 a day at RDU. I imagine it must be more at National, but it's got to be less than $40 a day! Or at least one would hope.