The correctly edited sentence is "Unusual and comical are the kinds of stories that she and I like best".
The change is in the pronouns her and me for she and I. All of them are pronouns, but her and me are object pronouns and she and I are subject ones.
Object pronouns usually come after a verb or preposition and they take the place of a noun that receives the action. For example, "look at me".
Personal pronouns are used in subject position. This is the case of this example: "That she and I like best" is a clause with a subject and a verb. Thus, the pronouns are functioning as subjects of the verb like.
