Your Main Quest isn't ruined. I think you were supposed to first talk with Burd, then ask anyone in Bruma about that second spy. Burt should only tell you that it's okay if you're going to kill somebody in the course of this quest, as you're a Blade and so everything you do is alright by definition. :rolleyes:
But never mind that; just go into her house now and ask her directly about her spying activities. Things should become quite straight forward from there on.