It looks like this is your current route: http://goo.gl/maps/dHwS7
I have extremely limited experience in that area, but have you tried going from 36th onto Tilden and using that to cross Rock Creek Park?http://goo.gl/maps/9nDBP Mind you, for the return trip, I’ve only gone up Tilden once, and though it’s a workout to overcome, you at least have a nice bike lane so you don’t have to bail onto the sidewalk.
In general, I like to stay in the street, even when on a steep hill and I’m going very slow, but that’s my preference. It depends on your comfort level and the area you are in (I generally avoid Connecticut Ave altogether plus in some areas like the DC Central Business District it is illegal to bike on the sidewalk).
I hope this helps and that you are able to find the most suitable route(s). I’m interested to see what is the best route around there.