To get to Kauai, you have to travel via Los Angeles, San Francisco or Honolulu.
Expedia suggests flying from Heathrow to Honolulu with United and taking the short hop across to Kauai with Mesa. For the dates, I entered, the cheapest fare offered was �656. (The total flight time is given as 22 hours 39 minutes. Flight times vary between about 19 and 30 hours, depending upon the route chosen).