I think that with cortana slowly going mad, she should try to access some foreunner database that includes a file that can destroy the didact, but you have to collect 3 index keys to access it, but just as you access it, cortana goes insane due to a virus created by the didact which speeds up t rampancy, and tries to kill you throughout the rest of the game. in the second to last mission, you attempt to destroy the core she resides in, and just before her death, she returns to normal briefly, and then chief and cortana share a sad final talk, then, in the last mission, you destroy the didact in a vengeful explosion filled finale.
