Ephemeral key is held constant in ElGamal
One thing Eve can do is compute the modular division of any two messages.

That is, she is given both $c_2 = mL^k$ and $c'_2 = m'L^k$; she can then compute the value $c_2\cdot c'^{-1}_2 = m\cdot m'^{-1}$.

Whether this, in itself, is sufficient to recover the messages depends rather on the distribution of Doug's messages.  However, what this means (independent of the distribution) is that if Eve has a guess for the plaintext for any of the messages, she can immediately deduce the plaintext for all the other messages.

Note that this similar system IES doesn't share this problem; if Doug repeats all the parameters, well, it will leak if he resends identical plaintexts; however it will still be secure otherwise.