Use ITA Matrix to give you plenty of options.
Here is one costing £767:
Itinerary
London (LHR) to Los Angeles (LAX) — Mon, Jun 15
London (LHR) to Frankfurt (FRA) — Mon, Jun 15
Lufthansa 903
Dep: 10:30 am
Arr: 1:05 pm
1h 35m
Airbus A321
Economy (K)
Snack or Brunch
Layover in FRA
1h 0m
Frankfurt (FRA) to Los Angeles (LAX) — Mon, Jun 15
Lufthansa 450
Dep: 2:05 pm
Arr: 4:35 pm
11h 30m
Boeing 747
Economy (K)
Meal
San Francisco (SFO) to Vancouver (YVR) — Mon, Jun 22
Air Canada 1843
Dep: 6:50 am
Arr: 9:07 am
2h 17m
Airbus A319
Economy (S)
Food for Purchase
OPERATED BY AIR CANADA ROUGE
Vancouver (YVR) to London (LHR) — Thu, Jul 9
Overnight flight
Air Canada 854
Dep: 6:30 pm
Arr: 11:40 am
9h 10m
Boeing 777
Economy (T)
Continental Breakfast, Meal
Fri, Jul 10
Mileage
11,702 total miles
Cost per passenger (including taxes & fees)
£766.68
Total cost for 1 passenger
£766.68
[...]
This ticket is non-refundable.
Changes to this ticket will incur a penalty fee.
Fare Construction (can be useful to travel agents)
LON LH X/FRA LH LAX 117.45KKNCQ33M /-SFO AC YVR Q7.50 154.00S21Q4TG AC LON 285.80TKXNCZGB NUC 564.75 END ROE 0.638555 XT 71.00GB 32.18UB 6.40DE 16.60RA 3.50YC 4.50XY 3.20XA 22.80US 3.60AY 14.30CA 11.10SQ 0.60XG 213.00YQ 2.90XF SFO4.50
Total Airfare & Taxes
£766.68