To avoid the muddy towpath, take Whites Ferry across to VA, then pedal just over 4 miles to get into Leesburg. It's an unpleasant ride on a busy route 15, but I believe it has a decent shoulder. But in Leesburg, you can pick up the W&OD paved trail which will basically take you all the way to DC. In Arlington, you branch off to the paved Custis Trail to take you to Rosslyn across the river from Georgetown. You can either cross Key Bridge into Georgetown, or continue on a trail on the VA side of the river to cross into DC over Memorial Bridge which takes you right to the Lincoln Memorial.