I am afraid you may need to be more specifically about "I would like to start dating you exclusively." She might take this to mean you won't date others but she can.
I would instead say, "I would like us to exclusively date one another."
Other than that it is a very nice note. I know that we generally aren't supposed to write them things, especially longish things that can be pursuing, but your situation is very different.
What is the time frame on the rental? 3 mos? 6 mos? Just curious. I wouldn't mention this to her.
I think you've done a nice job of appealing to the romantic that she seems to crave, and hinting at some boundaries without beating her up over the past.
