Stannis probably isn't dead. His last words to Brienne are "Go on, do your duty."
I think that might've triggered in Brienne's head that "Oh shit, that's right. My whole point in being here was to get Sansa and I let my vendetta get the better of me and lost my chance to sneak into Winterfell."
At which point she'll probably use Stannis as a bartering chip to get inside Winterfell's walls. Followed by her revelation that Sansa is no longer there. Brienne will ask for a couple of horses in return for Stannis and sets out to find Sansa before Ramsay does now that the word is out.
I think that might've triggered in Brienne's head that "Oh shit, that's right. My whole point in being here was to get Sansa and I let my vendetta get the better of me and lost my chance to sneak into Winterfell."
At which point she'll probably use Stannis as a bartering chip to get inside Winterfell's walls. Followed by her revelation that Sansa is no longer there. Brienne will ask for a couple of horses in return for Stannis and sets out to find Sansa before Ramsay does now that the word is out.