I'm directly between the two, and based on my cab fares to each I'd expect your cab fare to be $65, maybe a bit more.
theairportshuttle.com offers airport-to-airport transfers for $59 between the 3 area airports; that's a decent rate for BWI, but seems excessive for IAD-DCA.
supremeairportshuttle.com offers IAD-DCA for apparently $29.
Note that I have not used either and have heard no opinions about either.
If you were a week later, you'd have the option of helping to inaugurate the new Silver Line...