Oh don't give up - we have family in LA and friends in NY we often broke the journey in NY. You usually need to use an American airline so United or American and then they let you break the journey - if not free, then for not much more at all.
Call someone like Trailfinders or Travelbag, they are specialists. Can't remember whos normally cheaper though.
I find BA website can be really useful for working out which are the cheapest days to fly, then go elsewhere to see if they can beat the price on other airlines (like Thomas Cook). BA website when they are having their sales is hard to beat though.
Have used trail finders before. They can get you a cheaper flight by getting you on a stop over from somewhere else. My flight to NY from London has stopped over in London from new Delhi! Someone got off there so I had their seat the rest of the journey. I then hopped to LA from NY. Give them a ring
Do you have specific dates in mind or are you flexible?