Have you checked out the Holmes Run trail? That trail follows Eisenhower Ave (or merges with another trail to follow it), which goes through Hoffman Town Center towards Old Town with sharrows or bike lanes most of the way. Then once in Old Town meet up with the Mount Vernon Trail north (either Royal St or Union St while in Old Town), which takes you eventually to Memorial Bridge. There could be more direct routes, but that one is ~85% trail.
Alternatively if you can find a street route to Shirlington, you can get on the 4 Mile Run trail east to the Mount Vernon trail, and take that north.