Once you've decided where you want to be based in Los Angeles, travel search engines can help you find something in your budget range.
Kayak searches multiple consolidator sites and the hotels' own sites to find the best price on a room.
Some people find Kayak unwieldy because you have to go through a lot of steps to find the information you're looking for. You can also book directly through a travel search engine like Travelocity, Expedia, Orbitz, or Hotels.com. I have not found that any one of them consistently has better prices than the others on the same hotels, so I usually compare more than one.
Try Priceline. My brother Nardo recently shared with me his strategy on getting the hotel he wants at a discounted rate through Priceline. First he calls the hotel where he wants to stay to make sure they have an appropriate room available. Then he makes a Priceline bid for 35% of the reference price listed on Priceline, using the hotel's neighborhood as the location and the number of stars of that particular hotel. This works best at short notice. He has been known to use this technique from the parking lot of the hotel in question and get a room. You don't always get the exact hotel, but something close. Not all hotels participate in Priceline.
He also suggests that you start looking for just one night even if you need more. There's always the option to add nights if you get a hotel you like, and if you don't like the hotel, you can go somewhere else the next night. You can also try another search for more nights if your first offer is not accepted, since you can no longer immediately raise your bid without changing other criteria.
Rate-shopping tip from Rick Steves: Call. If you're not willing to risk Priceline putting you in a hotel other than the one you choose, there's another strategy that might work to get a reduced rate. Once you find the best rate you can on a third party site like Expedia or Travelocity, you can call the hotel and ask for the sales manager. Tell him or her the price you found online and ask if they will give you the room for 10% less. Since this is what the hotel would get paid by the discounter anyway, they will usually give you the same rate directly if you ask.
Additional Discounts: Additional discounts are available for members of AAA, Costco and certain credit card holders. You can also find hotel discounts in the Entertainment Book.

