Hi Hiro
As if he were reading the mind (that) Louise honestly didn't know if she still possessed, ...Louise is apparently asking herself if she has
lost her mind (i.e. lost her senses, gone crazy).
If you lose something, you no longer possess it. So, the sentence expresses "lost her mind" in an unusual way.
Louise wasn't sure that she still
possessed her mind. = Louise wasn't sure that she hadn't lost her mind.
The verb 'possess' is transitive in the sentence.