A better question might be, why did Daud try to kill Delilah? The answer, of course, is that the Outsider set him on the path towards Delilah, after which Daud's guilt led him towards protecting Emily.
Recall that the Outsider also appears to Delilah, presumably with some regularity, and we can guess that the Outsider has mentioned Daud's name to her. The Outsider does seem to enjoy helping along this sort of chaos so he can watch the story unfold.
Of course, it's also possible that, since Daud has been loudly running around Dunwall asking everyone in sight about a woman named Delilah, word got back to her and she just assumed that if a legendary assassin is looking for you, it can't be good. Killing Daud would then just be a smart precaution on her way to ruling the empire. (Even if you assume that word of Daud doesn't get out, Delilah could hardly fail to notice that anyone with any connection to her is disappearing.)
Or it may even be pettier than that. From Rumors and Sightings: Daud (found on Corvo's desk in the base game):
The earliest stories tell of a gang-killer without mercy, moving among the shop keepers and City Watch officers of Dunwall like a reaper through wheat. ... Daud is even purported to have spent a winter in the Academy of Natural Philosophy itself. And for a time, before a schism developed, he counted the Brigmore Witches among his allies. All the while, he honed his craft, and it is during this time that we believe he began to consort with the Outsider.
Unfortunately the game is not explicit on this point, so there is no canon answer. I assume that it is very much about self-defense, as there's no indication that she's going after Corvo, who is also touched by the Outsider; but there are valid alternate theories as well.