Are you able to test your Logitech folio keyboard with another compatible iPad? A visit to a local Apple iPad retailer or Apple Store will provide opportunity to test your keyboard with a different iPad.
If your Logitech keyboard is not working with any iPad, you have identified the source of the problem. However, if the Logitech keyboard works with other iPads - but not yours - you will then need to investigate the cause. If you are visiting an Apple Store, you can ask the technicians at the Genius Bar for assistance:
Genius Bar Reservation and Apple Support Options - Apple