Is the questioner sinful for her hatred and anger towards her grandmother, who insults people and talks about them behind their backs, even though she knows that her grandmother suffers from forgetfulness?

If the grandmother's insults and quarrels are due to senility and loss of mental faculties, then she is not legally responsible [in Islamic law]. However, if she is aware of what she is saying, then she has committed sins that necessitate repentance, because harming believers is a slander and a clear sin, and reviling a Muslim is فسوق (disobedience/impiety). You must advise her gently and kindly. If she accepts, then praise be to Allah. If she dislikes it, then you are only obligated to pray for her, and you can ask others to advise her. If she does not accept the advice, then you are only obligated to pray for her and seek forgiveness for her.