"Last season we finished second but it is not enough because we didn't win the Premier League", said the Argentine.
Vardy was unmarked to head Mahrez's fourth-minute free-kick into the far corner for his 19th league goal of the campaign. Lucas Moura won the ball high up the pitch and released Kane, who raced in on goal and shot past Eldin Jakupovic at the near post.
The win clinches third place for Tottenham in the table while Leicester stay ninth.
Only a brilliant stop from Hugo Lloris kept the deficit at one as the France goalkeeper produced a fine one-handed save to keep out Demarai Gray's effort.
It was all-square in the 53rd minute when Lamela's shot went in off Fuchs for an own goal and the comeback was complete seven minutes later with a carbon copy of Lamela's first - the Argentinian converting another Walker-Peters cross from close range.
Gerrard explains decision to take Rangers role
The Scottish striker has only been on loan at the club since January, making 17 appearances and scoring six goals. Cardiff City are reportedly aiming to make Rangers midfielder Josh Windass their first summer signing.
Spurs scored a second goal in a matter of minutes to make it 3-3.
"The number of players will be limited, like every season, and the market today is going up", Pochettino said, reports Goal.
Mauricio Pochettino will hold talks with chairman Daniel Levy next week in which he will outline a new vision for Tottenham and stress the need to take "risks" if they want to become a trophy-winning club.
Kane looked like extending Spurs' lead when he showed his threat in front of goal but was denied by Wes Morgan.
BC-SOC-Tottenham-Leicester, 438 Spurs leaves Wembley with 5-4 win vs Leicester, clinches 3rdAP Photo LON826, LON830, LON839, LON838, LON821, LON824, LON825, LON801, LON803, LON802Eds: APNewsNow.
Game over? No. Harry Kane had other ideas.
The hosts trailed 3-1 after 47 minutes, scored three goals in 11 minutes to lead 4-3, only to be pegged back to 4-4 before Harry Kane curled in the victor in the 76th minute.
The visitors, meanwhile, finish 9th.