Your client should buy it.
I'm betting that the license for AppGameKit specifically says that you may not provide the library and source code to anyone else.
Did your original contract with the client include providing the code for the game? If not, then you should negotiate a separate price for it. Once your client has the code, he can use it as an example for creating his own games. But, if that was part of the original contract, no sweat. He still needs to buy his own copy of AGK.
Cheers,
Ancient Lady
AGK Community Tester and AppGameKit Master