My brother used to rent a 2 BR in Culver City for $1850/month, which I would consider within 20 minutes of Beverly Hills (depends on exactly where you will be working - but even in peak times there are plenty of side streets to travel). Each area in your radius will have many (most) 2 BRs that go for more than your 2k price, but each one should have some options available. I think a lot will really depend on your expectations of a place (i.e. how modern does it need to be)? A good place to test out the waters is the apartments/housing section of losangeles.craigslist.org. Search 2 BR and then tab to westside/southbay. Ignore the southbay areas, but you can see what is available in West LA at that price and then map on google to start getting a feel for the area?

I recommend investing in a Westside Rentals account when you really start looking. I searched on craigslist for weeks when I needed to rent a house a few years ago. Then got a Westside Rentals account and found a place within days. It costs $60 for 60 days, but most people find places quickly - so you can buy their passwords on craigslist (for sale section) for $20.

I am sure you will get great information here. But would also check out the City Data Los Angeles forum as that is more geared towards moving and relocations.