Even if they don't, they may have some kind of information as to who might know. And it turns out they don’t Dragons aren't just coming back to life, they're being resurrected by other dragons
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Delphine suspects that the thalmor might be bringing dragons back for some sinister purpose
She's going to talk to her contacts about how we can infiltrate the thalmor embassy.
Once you tell her that the thalmor know nothing about the dragon attacks, she will be rather skeptical You can ask her why she doesn't believe you, and she will apologize. Not so much why she would believe the thalmor control the dragons, but there are political reasons to deploy them in skyrim if they could control them Per the treaty with the empire, the thalmor either cannot or at least should not directly intervene in the civil war.
When delphine asks you to kill him, it was written to create an ethical dilemma Logically a case can be made for killing him, but aesthetically it fails completely. She was directly involved in several of the most damaging operations carried out by the blades within the dominion She had been identified and was slated for the initial purge, but by bad luck was recalled to cyrodiil just before the outbreak of hostilities.
There, they find alduin, the same dragon that attacked helgen, resurrecting sahloknir, another dragon
I still think alduin being back for a while before helgen makes more sense, as it gives him time to begin resurrecting the dragons whose mounds delphine mentioned she found empty. Delphine thinks that the thalmor know something about, or are even responsible for, the dragons' return to skyrim She has a plan to get the dragonborn into the thalmor embassy, to gain access to their secret documents in an attempt to find out more about the situation at hand. I’m convinced delphine lied about the thalmor embassy mission
She tells you to break into the thalmor embassy and look for information about dragons She has no explanation or evidence that the thalmor know anything