You don't mention if you have an EIN. If you do, it should be a simple process of updating your tax info in your KDP account. If you don't have an EIN, you need to get one. It can be done in one phone call to the IRS. There are numerous threads from people about how to do this
If you do a search you will find several threads covering this. All you have to do is call the IRS. Can't remember the number off the top of my head, but they open at 7am, so call bang on 12/1 o'clock UK time (check the times, not sure what the difference is just now) and you stand a good chance of getting straight through. If you leave it later you can be in a queue for 30 mins plus which is expensive when calling the States. They will talk you through the EIN form and ask you the questions they need. It is better if you have your computer on and a copy of the form up on screen so that they can direct you to the questions they want you to answer. Takes about 10 mins then if everything is fine, they give you an EIN and you can update your KDP account.