Fans of both teams at Wembley paid tribute to Ehiogu, who died on Friday after suffering a cardiac arrest on Thursday at Spurs' training ground.
A minute's applause was held before kick-off and supporters at both ends chanted the name of Ehiogu, who had worked as Tottenham's Under-23s coach.
Mauricio Pochettino's men lost 4-2 to Chelsea as goals from substitute Eden Hazard and Nemanja Matic secured a dramatic victory.
Spurs had earlier twice come from behind to draw level, Kane and Dele Alli cancelling out Willian's brilliant free-kick and penalty.
Kane admitted Tottenham's players had hoped to win in Ehiogu's honour and said their ambitions would now switch to the Premier League.
"Of course the last couple of days have been tough," Kane said.
"Ugo was a great character around the training ground, it was just shocking news to be honest.
"Of course we'll do everything we can to win the league for him - we wanted to win today for him and for ourselves as well.
"There are still six tough games left in the Premier League. We can't control what Chelsea do now.
"We just have to win as many as we can and it starts Wednesday with a tough game away at Crystal Palace."
Tottenham dominated Chelsea for large spells but Willian's opening two goals were somewhat self-inflicted, with Toby Alderweireld conceding a free-kick on the edge of the box before Son Heung-min's rash challenge brought down Victor Moses in the penalty area.
Chelsea scored four goals from just four shots on target but Kane insists Spurs' authoritative display came as scant consolation.
"It's encouraging but obviously they've beaten us twice, we beat them once this season so that's what's disappointing," Kane said.
"We need to find a way to win these sorts of games. When we dominate large spells, Chelsea found a way to win and it was similar a couple of years ago in the cup final.
"We dominated that as well so it's disappointing. It's hard to take. We can't put our finger really on why, a bit of luck, a bit of momentum swung their way and it was just hard to come back from that."
Spurs were hoping to extend their own seven-match winning run while continuing Chelsea's recent dip, which had seen Antonio Conte's side lose two of their previous four games.
Instead, the Blues are likely to gain a psychological boost from their victory as they look to maintain a four-point lead in the title race.
Tottenham next travel to Crystal Palace on Wednesday, a day after Chelsea host Southampton.
"It's hard to say what effect it will have - of course we wanted to win and keep our momentum going but it didn't happen," Kane said.
"There's nothing we can do now. We can't control what Chelsea do, if it gives them more momentum, it gives them more momentum. We've just got to concentrate on our game.
"We've got six very difficult games home and away so we'll do everything we can to get as high up the league as we can."
