No - not mad. Yes - doable.

You should be out of Customs&Immigration in an hour or so.

Taxi IAD-DCA is metered and costs about $60, about 45 minutes
door-to-door.

Super Shuttle costs about half that, is door-to-door as well,
but could take longer than a taxi depending on passengers
loaded and stops to be made.

Taking the Washington Flyer Coach (every 30 minutes) or
the 5A Metrobus (every 60 minutes) and then MetroRail to
DCA will probably be iffy time-wise.

Note: if you are checking bags, be sure that they are properly
tagged at LHR so that they get to you in the Dulles IAB facility.
UA passengers arriving into IAD and connecting to UA domestic
ex-IAD go through a separate Customs&Immigration facility in
the C Concourse - and their checked bags are delivered to
them there. You are connecting to UA domestic, but NOT ex-IAD.