First of all, let me wish you a great and happy new year. When you get to LA, call me please, I do have a condo for 2 bedrooms that should be closing soon, and it will be ready for rental. It is in Encino, which is a great part of Los Angeles.
Your best bet is to look on craigslist.com, hotigloos.com or westsiderentals.com (which you have to pay for) or drive the area you want to be in. Be careful of scams...if they have a large 2 bedroom apartment for $800 a month, and they want you to send money to them upfront, these are scams. Or, when you get her, drive the area you are looking at and write down the phone number on the buildings and call them. This is possible to find. You will need to have good credit too and at least 2 months rent to put upfront, first months plus security deposit.
Good luck with your search.
All the best,
There may be pockets near Koreatown for that price, but I wouldn't expect much from the building or neighborhood. You'll have to look closer around the 10fwy or south of the 10fwy. Let me know if you need help with anything else.
$1500 is pretty low for the area. You might find something in the 90016 area. Craigslist has many scam artists on it so as the agent below suggested Westside Rentals is a good option.
The Carrabba Group