Nihongo ha kyuu kagetsu ni benkyou shiteiru
= I've been studying japanese for 9 months.
If it's wrong, why? Should it be "kyuu kagetsu go ni?"
I would rather say: 私は九ヶ月間、日本語を勉強しています。(Watashi ha kyuukagetsu kan nihongo wo benkyoushiteimasu).
In this case of a continued action in time you do not need at all the particle に after the word indicating how much time you have been doing something. Add instead 間 (kan) that indicates a span/length of time. Also, を (wo) should mark the object/what is that you are studying (in this case Japanese).
Actually, there is a related question here.