Hi Andrew and Welcome. My Name is ***** ***** I look forward to assisting you today with your question.
You can change the card details of your Amazon account from online (using your computer). Follow the steps below (from Amazon)
Go to Manage Your Content and Devices page -----> https://www.amazon.com/mycd
Click on Setting tab, select Edit Payment Method below Digital Payment Settings.
Select the credit card information, or add a new card, and then click Continue.
Enter the billing address, and then click Continue. You will be taken back to the Kindle Payment Settings page where you can view your edited 1-Click payment method.
When you change the card details from there, the card details on the Kindle will also be changed automatically.
If the changes doesn't reflect on your Kindle, then reset the Kindle and see if it works or not. Press and hold down (or slide) the Power button for full 20 seconds and this will reset the Kindle.
If that didn't work, then you will have to de-register the Kindle and re-register it. This will update the account with new card info. Go to Settings> device and/or Registration and de-register from there. Then re-register it from the same menu. That should fix it.
If you have any difficulty on following the steps above, please let me know and I will continue to help you.