Vox Mentis

Getting through immigration actually isn't that hard. The important part is limiting how much followup you'll expect to receive from immigration when your passport isn't shown to be leaving the country. While unlikely, it's possible that the passport you were given has a visa for your trip. You should probably confirm that this isn't the case first, since if you have one you should tailor your story to the type of visa you were given. Without a visa, you have two realistic reasons to visit - tourism and family.

If you're planning on assuming a new identity entirely, which is certainly a feasible option, it honestly doesn't matter. You could claim to be a tourist who forgot her return ticket, give a popular hotel name in a populated city, and then use a different name on the first person you meet outside the airport. The trouble with this route will be leaving. You'll need to get a new passport when it's time to return to Los Angeles or find another way of leaving the country illegally, which could pose a problem.

This leaves family. Visiting for a funeral would explain why you had to arrive on such short notice. If you claim to have relatives that are picking you up outside of the airport, you can even justifiably not know the exact address of where you're staying. Furthermore, you will have at least a few months before your stay becomes questionable and you need to respond to inquiries about why you haven't left the country. By then, you'll be a bit more established and can decide if it's worth remaining as Emily and coming up with an excuse for your extended stay or finding another way home.