If the input text and output text in DES are the same what is the value of the key? If some 64-bit message block (e.g. $M = \text{“12345678”}$) was encoded in DES using one key and the encoded message had the same value (i.e. $E_k(M) = \text{“12345678”}$), would it be possible to determine the DES key based on that?
I'm not aware of such DES key which $E_k(M)=M$

DES has 4 Weak Keys which allows $E_k(E_k(M))=M$ 


i.e. if you encrypt the encrypted message, you get as result the original message.



  page 159-160

Keys before parities drop (64 bits) | Actual key (56 bits)
0101 0101 0101 0101                   0000000 0000000
1F1F 1F1F 0E0E 0E0E                   0000000 FFFFFFF
E0E0 E0E0 F1F1 F1F1                   FFFFFFF 0000000
FEFE FEFE FEFE FEFE                   FFFFFFF FFFFFFF

  
  Example 6.8 Let us try the first weak key in Table 6.18 to encrypt a
  block two times. After two encryptions with the same key the original
  plaintext block is created. Note that we have used the encryption
  algorithm two times, not one encryption followed by another
  decryption.

Key: 0x0101010101010101
Plaintext: 0x1234567887654321 Ciphertext: 0x814FE938589154F7

Key: 0x0101010101010101
Plaintext: 0x814FE938589154F7 Ciphertext: 0x1234567887654321