Block cipher fixed points (plaintext equal to ciphertext)
The question makes a number of statements that are incorrect.  It is not correct that a fixed point is guaranteed to exist.  It is not correct that if you hold the plaintext constant and vary the key, then a fixed point is guaranteed to exist.  Moreover, the existence of fixed points has only an extremely tenuous connection to security.

Assume $E$ is a secure block cipher.  A reminder of standard notation: $E_k(x)$ denotes the encryption of plaintext $x$ under key $k$ using this block cipher.

Here are some results on the number of fixed points, for different mappings derived from the block cipher $E$:


If we fix a key $k$ and look at the mapping $x \mapsto E_k(x)$, then we expect this to look like a random permutation.  The expected number of fixed points in a random permutation is 1, but this number varies from permutation to permutation.  The probability that a random permutation has no fixed points is $1/e \approx 0.37$.  For more facts on fixed points of a random permutation, see Random permutation statistics.
If we fix a plaintext $x$ and look at the mapping $k \mapsto E_k(x)$, then for most modern block ciphers we'd expect this to look like a random function.  If the key has the same size as the ciphertext, then the expected number of fixed points (treating it as a random function) is again 1.  The probability that a random function has no fixed points is again $1/e \approx 0.37$.


See also Does AES have any fixed-points?, and also The Ideal Cipher Model (wonky) on Matt Green's excellent blog.

The reason a longer key provides extra security has nothing to do with fixed points.  It has to do with the fact that a longer key provides better resistance against certain other attacks (e.g., brute-force keysearch).

In short: I think you are laboring under some misconceptions about the existence of fixed points in block ciphers, and their relevance to security.