In nine days, I would go to a total of three cities in two countries. Of course it takes more time to "fully appreciate" a place but 72 hours for each isn't that bad (provided that you want to explore them a bit and are not here just for relaxing). Also, you'll want to get around a bit and spread the risk that you don't especially like a place (there is a ton of good places in Europe but preferences vary).
Anyway, I'd try to make it a 14 days trip, with a slower pace you still could comfortably go to four different places.
On the other hand, our replies here tend to propose a city tour and you might've thought about a more adventuresome vacation in terms of never sleeping twice at the same place... That's of course a possibility, too, but as a general rule requires a bit more prearrangement or money.
Denmark, Sweden, Germany and Switzerland are great destinations, Italy too... Italy and Germany are a bit cheaper for bread-and-butter products and except maybe in Italy (depending on your flexibility), English will do the job.
Make your decision based on internet/travel guide informations or on a gut level and drop me a PM if you plan to go to Switzerland and still remember this thread and if you like.
(The Canadian flag remarks are not meant to be taken seriosly, most Europeans aren't prejudiced against Americans and who should make sure that doesn't change if not travellers?)