I don't understand why I have 11 rounds? What is the role of the 11th round?
I suspect that they are doing standard AES-128, and thus have 10 rounds.
By '11 rounds', I suspect they mean '11 subkeys'; AES uses one more subkey than it has rounds.
Let's suppose that I want t attack the 10th round, must I compute the 10th ciphertext?
I rather suspect what's actually needed to perform the attack will depend on the details of the attack. On the other hand, without the final subkey, you can't compute the intermediate cipherstate (and if you knew the final subkey, that'd give you the entire key anyways).
I suspect that it's easiest, for a DPA-style attack, to attack either the first or the last subkey; those subkey bits satisfy the equation $S = K \oplus I$, where $S$ is one of the subkey bits, and $K$ is a bit you know (it's either a plaintext bit or a ciphertext bit), and $I$ is a bit of the AES internal state; if you can find a correlation with $I$, you can recover $S$; recover enough $S$ bits, and that's the game...