The Kurt Gödel papers were donated by his wife to the Institute for Advanced Study in his memory. The web page of the archive can be found here.
The Kurt Gödel Society maintains a library containing about 30 books as well as the Journal of History and Philosophy of Logic. The library is accommodated at the library of the Department of Mathematics of the University of Vienna.
The speech held by Hilbert at the meeting in 1930 in Königsberg where Gödel first announced his incompleteness theorems, as well as the four minutes excerpt that was broadcast and has been preserved, can be found at the web page of James T. Smith.