This.
She won't continue unless you look like a chef. Meaning, you must wear the stupid hat. lol Sometimes assassins must work in disguise, this is one of the weaker examples but still necessary.
If she won't continue conversation after you have the Chef's hat on, then I'd try reloading from an earlier save. I personally have never had any problem with any DB quest glitching/bugging, but other quests have been a simple matter of reloading a save and letting the Xbox re-think that particular part of a quest line. This usually gets me past any difficulties that I'm having.

Yea I am sort of a role player in some areas, although not so much as a lot of others to eat every day and sleep every night... but my assassin would find and change into different outfits to serve the role I was playing. Like changing into a couriers outfit (stormcloak/imperial) that he was wearing to deliver his message. I also bought an entire chefs outfit complete with shoes before even starting that part of the quest. My assassin relied almost entirely on distance and surprise to get the job done, so I never needed to wear armor. It always made me laugh how much you could look like an assassin, and people wouldn't treat you any differently than if you were wearing street clothes.
If you were on the run from a powerful person, would you let a hooded man in all black wielding a dagger as close to you as a wounded hunter wearing only a bow and some bloodied furs? A good assassin is part con man part hit man, but in this game the name of the game is play how you want to play... so everyone is completely oblivious to who you are, what you say, who you swear allegiance to, what you wear, and what race you are. Skyrims a good game, but there's a reason why people who could play morrowind up until the day oblivion came out are bored just a month or so after skyrim came out. You only have to do the quest once and SOMETIMES twice to see all possible outcomes. No quest seems to have more than 2 outcomes or ways of doing it, most questlines only ask you once to choose between a path during it.