You just acted on your own expection rather than an offer letter. The question is why you did not tell your friend your salary expectation at the first place, nobody can read your mind!!! Obviously, the miscommunication is on your side, particularly the " 一直沒好意思問" at the beginning (why) and now 打算寫信給她所在學校的HR和係裏的Dean把這個事情說道說道, My suggestion is not to do that, it will not get you the job and make you look real stupid.