A literal response to satire

The only way of disambiguating satire or any figurative language (and most language is figurative) is shared knowledge of the world (and culture). The only way for Jim to know that Jane is speaking figuratively is that he knows she actually does not have a .45 or that she is extremely unlikely to have one. Or possibly, assuming Jane is on duty police, having a .45 is in no way relevant to the playground. You also have to start with the assumption that there is some real world context in which Jane could have interpreted Jim's statement as non-hyperbolic (e.g. the playground is a known place for drug dealing).

The assumption that there is some universal procedure for identifying figurative language is behind much of the unjustified British conceit that Americans don't understand irony. But this is simply caused by a lack of shared cultural (world) knowledge. Both Americans and Brits use irony/satire in about equal measure but their usage spheres do not overlap perfectly. The misunderstanding goes both ways but the stereotype developed only in one direction for other reasons.

However, this misunderstanding is not only cross-cultural. In any interpersonal relationship, the overlap in shared knowledge of the world (both general and immediate) is imperfect, thus leading to many possible confusions. Language (communication) has many conversation repair instruments such as 'just kidding' but that does not always prevent issues since figurative language (or which irony or satire are just one small part) often evokes very emotional responses.