If we can use trust law as our starting point (trust = shared title to property), then we see that an Amazon/Kindle account is just a trust between you, amazon, and this other party. The only one investing money in the trust is this third party.

If you delete the account, you have two of the three parties' agreements documented. You need all three, in order to be able to claim that nobody's rights are being infringed. This can be done by making a good faith effort to notice the third party via multiple forms of communication. Even if they don't get to the party, you need to at least make the attempt. Send a first notice, wait ten days, if no response, send a second notice. If no response, consider that your acquiescence, and delete the account.