Where should I put my mirror?

Pyth - 19 17 15 13 bytes

Thanks to @FryAmTheEggman for saving me two bytes.

ARRGH the special case for no answer. Solved that!

e_I#jL\|cL2.:

Test Suite.

e                Last element in list, this works out to longest one
  _I             Invariance under reverse, this detect palindrome
   #             Filter
   jL            Map join
    \|           By "|"
    cL2          Map chop in two pieces
     .:Q)        Substrings. Implicit Q). ) makes it do all substrings.

05AB1E, 19 17 14 bytes

Code:

Œévy2ä'|ý©ÂQi®

Explanation:

Π               # Get all substrings of the input
 é               # Sort by length (shortest first)
  vy             # For each element...
    2ä           # Split into two pieces
      '|ý        # Join by "|"
         ©       # Copy this into the register
          Â      # Bifurcate, pushing a and reversed a
           Q     # Check if it's a palindrome
            i®   # If so, push that string again
                 # Implicit, the top element is outputted

Uses the CP-1252 encoding. Try it online!.

Python 2, 102 97 bytes

def f(s):h=len(s)/2;r=s[:h]+'|'+s[h:];return s and max(r*(r==r[::-1]),f(s[1:]),f(s[:-1]),key=len)

Rather slow and inefficient... Verify the smaller test cases on Ideone.