Kane is two goals ahead of Everton's Romelu Lukaku heading into the final weekend of the season following the rout at the King Power Stadium.
Shakespeare also challenged his players to make amends by ending the season with victory over Bournemouth, which would allow them to finish in the top half of the table.
Son then put daylight between the sides with a low effort which skidded across goal beating Kasper Schmiechel in the 71st minute.
Tottenham's Harry Kane was all smiles following his four-goal performance against Leicester City. Kane had not finished when he scored again in stoppage time, with nearly a carbon copy of his third, beating Schmeichel low to his right from the edge of the area.
Pochettino was visibly frustrated when Son blasted a golden chance over the bar after Dele Alli blocked the ball into his path, but the South Korea forward was decisive in helping Spurs take the lead after 25 minutes.
Leicester City: Schmeichel; Simpson, Benalouane (Gray, min. 66), Fuchs, Chilwell; Mahrez, Ndidi, Amartey, Albrighton; Okazaki (Slimani, min. 46), Vardy (Musa, min. 79).More news: What Is Ransomware? How Can Internet Users Protect Themselves From It?
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"We spoke a lot about changing that feeling and finishing in the best way", he said.
With 59 minutes gone they found a lifeline. Spurs' keeper, Lloris, had rushed out of his box to clear the ball.
Hope of a comeback lasted barely four minutes, however. "Now I'm in the driving seat but I'm not resting on my laurels". It is a season again where he fights to be the top scorer in the Premier League.
Kane has attracted interest from Manchester United in the past, with reports cited by the Manchester Evening News in March suggesting that Jose Mourinho had identified the 23-year-old as a back-up target in the event that he fails to sign Antoine Griezmann from Atletico Madrid.
Kane has missed eight league games with injury this season but looks set to finish as top scorer, unless Everton striker Lukaku or Arsenal forward Sanchez have a remarkable final day of the season. It ended; Leicester 1 - 6 Tottenham. To score more than 30 goals (Kane has 32 in 38 games overall) is a big achievement.
Kane opened the scoring before Son made it 2-0 at half-time and, although Chilwell pulled a second-half goal back, Spurs scored four times in the last 27 minutes, including Kane netting twice at the death. But for his well-documented injuries he might well have reached 100 Premier League goals already, but Sturridge is still only 27 and will be in demand if he moves on from Liverpool this summer.