Best steps are

1. Take daylight flight to LHR
2. In a week or two before the flight, mimic the new time zone as much as possible. That would mean people in North America going to bed early and getting up early.
3. If you can fall asleep easier in business class, consider business class.

The rest, like diet and pills, are way down the list as far as effectiveness.

Once you arrive, nap if tired but make it a short nap if it is daytime. Try to sleep during the night. If the trip is short, try to mimic the time zone in the US, which means getting up late in the morning, as late as possible on the morning.