Calculating ECDSA m or k (k or d) when s1 = s2
However, I'm confused by what I should do with two different messages ($e_1 \ne e_2$) with the same $s_1$ and $s_2$

Well, if we consider how $s$ is computed:
$$s_i = m^{-1} ( \text{hash}(e_i) + r \cdot k )$$
If $s_1 = s_2$, then (because the private key $k$ is the same in both cases, and $m$ (the secret nonce) and $r$ are assumed to be the same, we have $\text{hash}(e_1) = \text{hash}(e_2)$
If $e_1 \ne e_2$, that means that we have a hash collision on our hands; because the signature depends on the hash of the message (and nothing else about the message), that means that (as far as ECDSA is concerned) we're signing the same message twice, and so the attack isn't possible.