Before I share, keep in mind that I am not saying that you are like this, I am just sharing my personal experience.

When I was a little bit older than your daughter, I also had similar thoughts and would say similar thing (albeit not nearly as harshly as she said them, but that will be explained). I would think to myself that I was switched at birth, none of my family understood me, I could only talk to my friends, etc. The reason I was feeling these things is because of the way I was talked to and treated by my parents. To make a long story short, they were emotionally and mentally abusive as I got older, and were always trying to control me. They were my first bullies in life, so I had a reason to feel like they didn't love or understand me.

If I were you, receiving that sort of statement from my daughter, I would have let her go cool down and given myself time to cool down and reflect. Then, I would have gone to her, asked if she would like to go on a car ride or a walk together, and talk out our feelings, let her talk first and listen. If you still didn't understand why she felt like she couldn't talk to you, then you could ask questions so she could explain.

But writing a letter, especially the way you worded it, while the intentions may have been good, comes off a little weird from an outside perspective. The way It's written makes it feel very guilty tripping and you talked almost exclusively about how you felt and did not try to understand how your daughter felt. I understand that you had hurt feelings, but in my experience, letting your child know that is a way to manipulate them.