Why does the key for a one-time pad have to be uniformly distributed? I would like an intuitive argument for what goes wrong in the proof that that a one-time pad provides perfect secrecy, if the key $K$ is not chosen uniformly at random from the entire key space.
Intuitively, each key decrypts the ciphertext to a different plaintext.  So if some keys are more likely than others, then some plaintexts will also be more likely than others.  Thus, unless all possible keys (and thus all possible plaintexts) are equally likely, observing the ciphertext will reveal information about the plaintext.

(Note that I'm implicitly assuming above that the attacker has no prior knowledge of what the plaintext might be before observing the ciphertext.  That's fine for a counterexample, since a cipher that provides perfect secrecy must by definition reveal no new information about the plaintext to any attacker.  So if we want to show that a cipher does not provide perfect secrecy, it's enough to show that it can reveal some information about the plaintext to some attacker.)