The Penguin's Nest (1) - S2-E27
Trivia: The guest cameo in this episode is by Ted "Lurch" Cassidy, who appears at the window as Batman and Robin climb up the side of the building. Batman advises him to go back to playing his harpsichord - which has been regaling us with the theme from The Addams Family since the scene began.
Answer: There wasn't one singular city used. It was shot in and around various locations in California, plus on a lot of sets and backlots. Ex. Some scenes were shot in LA, some in Pasadena, some in Santa Barbara, etc.
TedStixon