EVERYTHING in LA is expensive. It's LA. If you think LA has expensive hotels, try Monterey. They're selling cheap, Motel 6-style rooms, for over $250 per night. Sheesh.
Since LA is so spread out, there is no single place that is close to EVERYTHING. Most people who come here tend to stay in Hollywood, Santa Monica, or Anaheim.
In Hollywood, I recommend the Renaissance Hotel at Hollywood and Highland, the Best Western Sunset Plaza, or the Holiday Inn Walk of Fame.
In Santa Monica, the Holiday Inn Santa Monica is close to the pier and reasonably priced.
In Anaheim, I recommend the Howard Johnson (HoJo) Anaheim, and the Candy Cane Inn.
I would be really surprised if you found a cheap hotel in Westwood. It's an expensive area. Check tripadvisor before booking. Some of those places are real roach traps.
Westwood would be nice if it's a decent hotel.