That goal saw Kane equal Sheringham's tally and the England striker broke it 12 minutes later when he prodded the ball past Jordan Pickford from close range.
The former Bayer Leverkusen man said he wants to show better performance for his London club.
Inexplicably, Son is one of the most under-rated stars in English football and even at Spurs, he is not one of the club's A-listers, lagging behind Kane, Dele Alli, Christian Eriksen and keeper Hugo Lloris.
Son Heung-min opened the scoring for Tottenham at Wembley with his fifth consecutive home league goal - the first Spurs player to do so since Jermain Defoe back in 2004 - before Kane doubled their lead moments after the interval.
Klopp says Coutinho replacement needs to be correct player
Roberto is known for his flexibility and is fit for playing in the two positions which is something that Klopp is searching for. It is not that they say "You've got this but on the other hand we've done this and this and so you can not spend".
This season has already seen Kane break Alan Shearer's record for most Premier League goals in a calendar year with 39, while also finishing 2017 with more goals than Lionel Messi and Cristiano Ronaldo.
Spurs continued their fine recent form as they earned a precious three points that puts them joint fourth in the Premier League table with Liverpool.
But the Toffees were cut to ribbons four minutes later as Aurier, Son and Alli all combined with one-touch passing to set up Eriksen, who drilled a first-time shot past Pickford from Alli's deft flick. I didn't expect to see it.
Everton had the ball in the net eight minutes later, but it was ruled out for offside.
In this photo taken by the EPA, Tottenham Hotspur's Son Heung-min (L) celebrates after scoring a goal against Everton during the English Premier League match between Tottenham Hotspur and Everton at Wembley Stadium in London on January 13, 2018.
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The South Korea forward has been involved in more goals at Wembley this season than any other Tottenham player, scoring seven and assisting four.
"It was a shock second-half performance".
62' - Dominic Calvert-Lewin replaces Cenk Tosun for Everton.
- This was Tottenham's biggest margin of victory in a game against Everton since a 10-4 win in October 1958. I saw some good signs of the players last week against Liverpool and I saw some of the worst signs in the second half today which is massively worrying.
Allardyce's side are now winless in their last five Premier League games and have lost their last three.
'I never said sh*thole, though I did use tough language,' says Trump
The ANC is not in a position to stop any president from any country from saying anything they wish, she said. The remarks, Durbin said, were "vile, hate-filled and clearly racial in their content".