...aaaaand, lets start by getting honest with ourselves...
We would be looking at 13-14 nights...
I only say this because in almost every post that begins like this, there's usually some willful self-deception going on. Forget how many days you are taking off from work. Forget how many days you will be away from home. What counts - and what you should be counting - is the number of full days you have in Europe, not including your arrival or departure days. Those are not usable days. It takes about a day to get from North America to Europe. It usually involves flying overnight and arriving at your destination sometime around mid-day - if you have a connection or two, your arrival in your destination may be later. In terms of trip planning, it does not matter at all when you leave your home. What matters is when you get to your actual destination.
So, start by figuring out when you will show up at your first hotel - that's "Day Zero". Your trip actually begins the next day, after drag your jet-lagged self out of bed on Day One (you will probably not be functioning at 100% for that first day, so many of us recommend an easy first day, but let's ignore that for the moment). Next, figure out the day you fly home. That's not the last day of your trip (not in terms of it being a usable day). That's the day after your trip, or the day you spend getting home - you're not going to squeeze in another museum or cathedral that morning before you zoom off to the airport in the afternoon, the day is consumed with getting to the airport and on the plane.
So, begin with being honest with yourself about how many days you really have in Europe - full days, not including your arrival or departure days. Count the days you have, not the nights (yes, you can track where you will be each night, but you will do most things during the daytime hours). Only when you have this realistic number can you begin slicing the trip into X days here, Y days there, etc. And of course, be sure to account for the fact that simply the act of moving yourself from one place to another generally consumes a major portion of a day - even if you're good at it, if you're only going a short distance using efficient transportation, getting packed up, checked out, to the station/etc., the actual transportation, maybe getting lost once or twice, finding your next hotel, stopping for a meal, etc. - the basic logistics add up and eat anywhere from a half to an entire day.
July in Italy will be hot, maybe surprisingly so. Take that into account, too.
Hope some of this is helpful. Good luck!