"London, Bath, Edinburgh, Dublin, Paris, Bordeaux, Geneva, Zurich, Munich, Frankfurt, Hamburg, Amsterdam, Berlin, Prague, Vienna, Copenhagen, Arhus, Stockholm, Oslo, Reykjavik."
Dublin, Frankfurt, Geneva and Zurich can all be axed, in my opinion - they're generally a bit dull. I'd spend more time in some smaller more scenic places instead.
1.) Near Frankfurt, look into Rothenburg (touristy but well-preserved medieval city), and the Rhine/Mosel (think wine and castles).
2.)In Switzerland, head to the Bernese Oberland (near Interlaken, Lauterbrunnen). The natural wonders here are spectacular.
3.)Instead of Dublin, spend some time in Bruges, Belgium (maybe the most beautiful old city in Europe??, lovely beer, the Venice of northern Europe) once you hit the continent.
"And probably worth mentioning: As ridiculous as this sounds, I'm to a point where I'm close to scratching off doing any travel in both Spain and Italy being a solo female traveler. Nearly every person I've talked to (who has been there, not inexperienced travelers) has told me NOT to visit these countries for fear of my safety being a solo traveler. However, if you can tell me I'd be fine (which I would assume, but after so many tell me not to visit, I've taken the "ugh, fine, I'll go to those countries another time!" -route)... let me know!"
If you were with a friend, I'd be more inclined to spend time in the south. But you're not, and it can be incredibly, unbearably hot and humid south of the Alps on a regular basis - not so great with a backpack. I'd stick with the north and see the south in January sometime.