If it was one of those washes where they have the aggressive sprays
from underneath it may have gotten water into something. Why it would
go a distance and then quit works against my theory.

I'm not sure I understood your post.  Are you saying you washed inside
the engine compartment?  If so, and if this car has a distributor and
distributor cap, what I used to see happen a lot was a warm/hot engine
would get washed, the engine would get started up but not run for very
long (your 2 km trip), then shut off, and then the car would not
start.  What happens is that all the water that was on the engine
turns to steam and some of it gets inside the distributor where it
condenses back into water.  Then the car no run.  I've found  a lot of
water inside some distributors from that.  I'm pretty sure it wasn't
water that was just sprayed in or the car would never have started in
the first place, that's why I think it was condensed steam.

The fix is to take the distributor cap off, the rotor off, and dry
everything inside it.
--
Elbridge Gerry, of Massachusetts:

"What, sir, is the use of militia?  It is to prevent the
establishment of a standing army, the bane of liberty.  . .
Whenever Government means to invade the rights and liberties of
the people, they always attempt to destroy the militia, in order
to raise a standing army upon its ruins."  -- Debate, U.S.  House
of Representatives, August 17, 1789  

Battery "seems" to be running ok ???

I would pull it and get it tested on a load tester. Autozone will test it. A
weak battery can run the radio, and even the headlights, but still not have
enough power to turn the motor over.