Betsy - some different advice, based only on our travels with kids.
First, summer next year is a LONG way out for a 13 year old girl - way too far out for her to get excited. She knows you are going, so stop there, for now.
Second, on most of our trips, and definitely when the kids were 13ish, I involved them (and husband too) very little in the initial planning. This works for us. I know what we like - ruins more than palaces, small/strange museums or odd things (Ice man in Bolzano), climbing to the tops of cathedrals and counting the steps, etc - and can get a good general plan. I do ask for opinions on specific things "would you rather spend the extra day in Rome or Venice?"
But we also don't plan ahead for each and every day. I plan the NUMBER of days we'll be in a city but not a daily itinerary, except for arrival day. After that I take the guidebook to dinner each night and we plan what to do the next day. At this point EVERYONE chimes in because they are there, they are excited, maybe we walked past something earlier and they want to return. I'm not dragging them around and they are happy to participate.
Other tips: Definitely alternate museums/cities with other things - boat ride on Lake Como, bike tour out of Florence, etc.
Great idea to take a cousin, but I'd be hesitant to take anyone other than the closest of family friends (i.e. goddaughter). It's tough being 3 - always ends up being 2 against 1. Consider times to "divide & conquer". Let her sleep in some days while you and husband do the early touring thing, then switch it up later in a different 2/1 grouping. Let her wander a bit on her own, especially in the smaller towns where it is easy to figure out where you are.
Plan for downtime-back to the hotel in the afternoon to rest, read comics, etc.
Finally, she may just be at the really negative stage that my daughter & friends went through. Backing off now may give her time to grow up -even 6 months can make the difference.