OP--don't let the "DC" (ok, DC-area) nancies get you down. velogirl's on the tip and spot-on: there are lots of ways to get to where you're going. DC is a very block-to-block city in terms of sketchyness, and I have full confidence you'll be able to suss out a route you feel comfortable with, provided you get a good bike map (the ADC is really the only one I can think of).
if it helps, I commute in the dark, on trails, through gangland PG county, and have been doing so every work day, rain, shine, or snow, for 3 years. The only problem I've ever had was with a gangsta goose. Also regularly bike from upper Georgia Ave. to downtown. No probs yet except pollution.
I agree that rock creek park is a big (albeit scenic) detour. There are plenty of other routes that are fine. Just looking at my bike map, I see Alexandria to George Mason Bridge to the Mall, up 6th to 4th through Howard U. and around McMillan reservoir to the hospital--looks like a capital ride. 5 bros. burgers on the way if you take a little detour to Georgia Ave. at Howard U.
U St. and Columbia Heights are *very* cyclist-heavy neighborhoods just west and south of the hospital; there are a few streets that are sketchy close by, but they suck for biking anyways (i.e.: kenyon, irving, etc.), and I think you'll be able to suss out their suitability at 5 AM. They're easily bypassed by coming up from the South. I've seen a fully loaded fluorescent elderly gent commuter turn west from Michigan into those streets on his regular route, so it can't be that bad.
I personally despise upper North Capitol, especially the deviation at the filter beds/army retirement home. Super fast/no shoulder/lots of stupid drivers.