Thanks for the suggestions
The third time I tried to contact them, looking for an exact phone number to call, a chat option popped up. Using the chat, I found that they cannot send me a correct invoice by either mail or e-mail. However, the customer support person suggested that I should make a new order. I did that, he reduced the price of the Kindle by the amount of the discount that I got with the previous Kindle order and I'll be asking the UPS to return the Kindle back to Amazon on the next business day (tomorrow is a national holiday).
It's not a very practical solution, but it seems to have worked out fine this time.
I just hope there won't be an issue with the new shipment :P