Are you crazy? Or is your gf crazy? Sounds like it.
I backpacked around Europe with two other friends in 2002, and we took 45 days to go to 10 countries by train, and THAT was insane. No, seriously. We spent 1, maybe 2 nights in each city, most of the time staying with local friends I had (from studying in Iceland the year before). Even with the personal touch, it was an extremely demanding schedule and I would not recommend it for first time travelers. It will be a royal waste of your money and time, and you won't remember anything clearly.
For me personally, I had been around Europe twice before, so 1-2 nights was enough for me... the big European cities get boring fast, after you've done the tourist thing once. My friends had never left the US before, though, so they wanted to see everything all at once and got exhausted very quickly. I know one of them really wishes we had done things differently, but she never spoke her mind and got dragged along in the end.
Pick a handful of countries (no more than 4-5), a big city and a small town in each, and slow down a bit. Europe isn't going anywhere... you can always go back for a second dose later.
EDIT: clothes for 13 days is nothing. I wore a single pair of zip-off traveling pants my entire 45 days in Europe. I washed them 1-2 times a week (quick dry, in the hostel). Bring your own travel soap. Bring 2-3 pairs of underwear, and wash them every few days as well. A couple of shirts, a jacket, and walking shoes... and you're set. No, really, that's all you need. If your gf intends to go clubbing every night or something, she has the wrong idea about this trip (13 DAYS?!). Dress like you're going camping and wear the same clothes every other day, basically.