Continuity mistake: We first see from the inside of the door (and therefore backwards) the instructions that Mr. Miyagi leaves for Daniel to paint the house. In the next shot we see the same note from the outside - from Daniel's viewpoint. Clearly they are two different notes. The left & right arrows switch directions, the drawings of the hands change, the position of Miyagi's signature changes, and the placement of some of the exclamation points changes.
Chosen answer: He didn't lie. That was the company she was supposed to be working for. However, the company went bankrupt and she lost her job. She showed up at the restaurant at the same time that the hostess quit and was hired immediately.