Aj - these last two posts don't sound like someone who "is detached". Detaching is not about the physical act. It's not like you're a space shuttle detaching the rocket boosters after launch. It's about emotional unraveling of your feelings from her actions. It's not something you do once, it's a process you go through and it takes TIME. There are ups and downs and it comes from this process of letting go of the emotions, of the feelings, of the control.
You ask about a letter. Why do you want to write a letter? What do you want to say? What benefit do you think it will have? What expectations are you attaching to it?
Regarding the work trip, yeah, they can be rough. It's isplating enough going through this. Adding a business trip on top of it where you're staying at a hotel adds even more. Just remember it's ok to feel sad and lonely. But they pass. Think about doing something awesome instead - then do it. That will help you feel better!