Ehiogu, who was Spurs' Under-23s coach, suffered a cardiac arrest at the club's training centre on Thursday.
Spurs, who last won the title in 1961, lost the FA Cup semi-final to Chelsea on Saturday and are four points behind them in the league with six games left.
"Ugo was a great character around the training ground," said Kane. "The last couple of days have been tough."
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Former Aston Villa and Middlesbrough defender Ehiogu won four caps for England and had been working at Tottenham since 2014.
"It was just shocking news," added 23-year-old England international Kane, who scored in the 4-2 defeat at Wembley.
"Of course we'll do everything we can to win the league for him - we wanted to win (on Saturday) 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."
Tottenham have won seven successive league games and visit Crystal Palace on Wednesday, a day after Chelsea host Southampton.
Tottenham striker Harry Kane says they want to win the Premier League for Ugo Ehiogu, who died on Friday aged 44.