How to transfer game content from one Kindle to another?

I received a Kindle Keyboard from my cousin because he got a Fire and didn't want his old Kindle anymore. So, I registered it on my account and then the case broke and I needed to order a replacement. So, I got the replacement and now I want to transfer the games that my cousin had purchased on the original Kindle to the replacement. I cannot figure out how to do it since I don't show the games purchased on my Amazon account. Shouldn't I be able to move the games from the Kindle to my Amazon and then sync back to my other Kindle? Or maybe archive the games and then pull them up on my new Kindle? I'm afraid to perform an archive because the process involves deleting the item first and I'm afraid if I do that I might never get it back again. Want to make sure first that this can be done. Any help would be greatly appreciated. I'm a Kindle newbie.