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 " 一直没好意思问" at the beginning (why) and now 打算写信给她所在学校的HR和系里的Dean把这个事情说道说道, My suggestion is not to do that, it will not get you the job and make you look real stupid.