Bayern are unbeaten under Kompany, scoring 30 goals in seven games, are top of the Bundesliga and beat Dinamo Zagreb 9-2 in their Champions League opener last month, with Kane scoring four.
Kompany replaced Thomas Tuchel in the summer after leaving Burnley, despite being relegated from the Premier League with the Clarets last season.
Bayer Leverkusen boss Xabi Alonso, Germany manager Julian Nagelsmann and Austria coach Ralf Rangnick all turned down the job.
But, on his return to England, Kompany believes he has nothing to prove.
The former Manchester City captain said: “It’s important to separate things. Every challenge is different and the context is different. The most important thing for a coach is to say ‘have you done everything you could? Have you done your best for the team?’
“If you go home every night and at the end of the day if have you done everything you could to make the team better then you should have no regrets.
“I don’t come here carrying anything from the past apart from lessons. It’s different players, clubs, it’s a different context, situation and season and you have to leave it behind and focus on what’s ahead.”