LF really overplayed his hand. He only had the vale army by order of Robin to retake Winterfell. Everyone from the Vale viewed LF as an interloper. They wouldn't kill LF because he had won favour with Robin, but they wouldn't rush to his defence either. So LF tries to win Sansa's allegiance by helping to take Winterfell, he fails at the former, yet remains in Winterfel. Standing in dark corners looking shifty. Then Arya shows up out of nowhere and suddenly LF has this plan turn her against Sansa. He knows next to nothing about her yet presumes that if he has suspicious meetings with people she will take notice, follow him, pick his lock to his room, and find a note hidden in a tiny hole in his bedding. This note, he believes, would cast no suspicion on himself and is enough to turn Arya against Sansa despite all the hardship the Starks have suffered. Somehow he thought this would win him favour and trust with Sansa, even though Sansa is at home in Winterfell and can basically trust everyone but LF.
It really didn't take Bran's powers to allow this plan to fail. LF got desperate and sloppy.