Gareth Southgate's men are among the favourites to win the tournament in Germany but Friday's farewell party at Wembley fell flat and the players were booed off in a half-empty stadium.
Iceland took an early lead through Jon Dagur Thorsteinsson and the visitors then had the better chances to extend their advantage as the hosts toiled at both ends of the pitch.
England captain Kane knows lessons have to be learned ahead of their Group C opener against Serbia in Gelsenkirchen on Sunday.
"Sometimes you need this wake-up call," he said. "There are areas we need to improve, especially with our pressure.
"The group games are going to be very similar to this. We are going to have to defend well, especially the longer balls, and with the ball we need to be a bit more patient."
Kane, 30, said despite the defeat there was a lot of justified optimism among England fans that the Euro 2020 finalists could go one better this year.
"Ultimately it is down to us to make sure we get it right on the pitch," he said.
"This could be a nice wake-up call for everyone to realise it is not going to be as easy as people think. There is a lot of work to do."
Kane, who scored 44 goals in 45 games in his debut season at Bayern Munich, said he was raring to go.
"Being so close in the last Euros, there is a real hunger to be back. It comes around quick, if I'm honest," he said.
"We had the World Cup and we are back here again. I feel like we are on the right path. But as we all know as well, it is really difficult to win these tournaments.
"There are a lot of things that you have to do right and prepare right, and you need a bit of luck here and there. Every winner does.
"We have that mindset that we can achieve it. We also know there is a lot of hard work from now until the final in July."
England's other opponents in Group C are Denmark and Slovenia.