There are is a lot of discussion in the bioethical/legal communities surrounding this issue – this is a good short summary, as well as this.
Property rights in the body generally, and organs specifically, are considered as a sort of “quasi property” right – it really depends on what the part is, if the person is living or dead, the use of the material, etc. Organ sale is forbidden under U.S. law – there was an interesting case where a guy put up organs for auction on e-bay. So, it’s fairly clear that regardless of the recipients knowledge of whether the organ was coming from that the entire transaction in the U.S. would be understood to be illegal.
Now, can you sue to get it back? You can – you’ll lose. It’s not going to matter whether the person will die or not without the kidney. A man in NY recently sued his wife to get his kidney back or the cash equivalent of its value. The concept itself was pretty offensive, which shows you where we are legally at this point as to suits for the original “owner” of organs that have been transplanted.
An interesting aspect of this, of course, is that an organ transplanted into the body of another is, in many ways, no longer yours. It is integrated into the system of the recipient, and is therefore going to contain, if examined or removed, tissue, fluids, etc. that are completely native to the recipient and alien to you. So part of the question is not only you’re right to the kidney, but also you’re right to the aspects of the kidney that reasonably belong to the recipient that they would have to give up and you never had.
On the money side, in this case, I bet the result at this point would be a big, big reward for you from all parties, jointly and severally. There would be a huge policy consideration in allowing a person to keep the kidney with impunity. The court would be encouraged to set precedent that would disincentivize the behavior – a big settlement might, “reasonably”, act to make people reconsider a transaction like this knowing that, if found, the consequences to them could be devastating.
Of course, I don’t think there’s anything that would have any effect that the court could do. A person in such grave need is acting on survival instincts – you can’t disincentivize survival.