I will probably deserve the flaming I get for this, but:
While investigating bringing my mother over, I found that KLM have a ~GBP1250 J-class fare from LHR/LCY to LAX via AMS. It's got a 7-day minimum, I think, and is available till 15 March or thereabouts. I turned it up on expedia.co.uk and KLM's own UK website.