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.
The South Korea global was involved again as Kane netted his first of the game two minutes after the restart.
Teddy Sheringham had previously held the record of 97 goals in England's top flight for the club, but Kane met and surpassed the tally with his double, tapping home early in the second half (47) before bobbling the record-breaking goal into the net 12 minutes later.
Chelsea were unable to pierce Leicester's defence and were second-best for long periods at Stamford Bridge, even after visiting left-back Ben Chilwell was sent off for a second bookable offence midway through the second half.
"The good thing is Jimmy Greaves' (all-time club record) is far away, so not every week he is going to break something". 89' - Wayne Rooney crosses for Dominic Calvert-Lewin but Jan Vertonghen gets just enough on the ball to divert it away from the Everton substitute.
On the day it was revealed Kevin Keegan had returned as manager, Emre was brought back into the team for the cup replay against Stoke and with Newcastle one goal up, managed to get himself sent off after 29 minutes.
It was Kane's 98th goal for Spurs in the Premier League making him the club's all time top scorer in the competition. In doing so, Son joined Jermain Defoe as the only Tottenham player to score a goal in five consecutive home matches.
Yet Everton's hopes of reeling in Tottenham lay in tatters two minutes after the restart. "It went my way so I'm happy".
It hardly mattered, however, as Kane extended Spurs' lead moments later with a close-range finish over Pickford from Eric Dier's precise pass into the box.
And expect him to end his 2018 Premier League scoring drought just 13 days into the new year by shooting down Sam Allardyce's Toffees.
"I'm 100 per cent confident", he said.