Don’t know exactly your step-by-step installation process. It sounds like you haven’t exactly installed the plugin. Maybe something with your installation file, although you said you already uninstalled and reinstalled. It depends if you install directly into your WP through front end (click of button install) or back end (manual loading into wp-content/plugins).
Not to complicate matters, I would suggest you really verify the plugins file (complete and not corrupted), because your error says “Error Details: Jetpack ID is empty. Do not publicly post this error message!”
