Lu Lu's letter says Billy Baxter was giving the party, but the emcee at the party introduces Michael as the host. Later, at the party, Jack (disguised as Orizon), cons Michael out of his cash, but the subsequent title card says Jack "borrowed" the money from Billy Baxter. The property crew apparently got confused about which character was which. Billy was merely Michael's friend who was holding the stakes of the bet between Jack and Michael.
After the first week of the bet passes, Jack receives a letter from a hotel regarding the discontinuing of his room and credit. However, it had already been established that Jack had been sleeping in parks that entire week.
When Jack realizes he lost Lu Lu's phone number, she had just gone into her hotel a few seconds earlier. Considering how very anxious he was to connect with her, Jack should've had no qualms about simply following Lu Lu inside and asking for a replacement card, especially since the film later implies that Jack was also staying at that hotel.
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