The FA was also sanctioned due to "disturbances during the national anthems and setting of fireworks" by England supporters at Wembley, some of whom booed the Danish anthem.
The experienced businesswoman is set to join the English FA in January, 14 months after Greg Clarke resigned in fallout from making offensive comments at a parliamentary meeting.
Kane's brace of goals had enabled England to edge past a gritty Tunisia in its opening match at the FIFA World Cup in Russia.
The appeal was rejected despite FA support as the paperwork was submitted 14 seconds late on deadline day.
The English Football Association has charged Sunderland manager David Moyes for his alleged offensive comments to a female reporter.