The Premier League champions reached the quarter-finals of the competition at the first attempt but a disappointing domestic campaign means there will be no continental football for them next season.
Schmeichel thinks the club could again qualify for a place among the European elite in future years but at the moment is focused on finishing as high in the league as possible, not the reports linking him with a move to Manchester United to follow in the footsteps of his father, Peter, should David De Gea leave for Real Madrid this summer.
"Those Champions league nights are special," Schmeichel told the BBC. "There is something about the night games, going away to different cities, it makes it unique.
"It's something I hoped would come earlier in my career. The journey I've taken has made me appreciate it even more - but also want it even more.
"Eighth position is still a real possibility for us with the games remaining and we will be aiming for it."
Leicester, who were flirting with relegation earlier this season, have clawed back up to 11th in the standings since Craig Shakespeare took charge from Claudio Ranieri in February.
Schmeichel believes the players let down their former manager and that the turnaround that has seen the club pick up six victories in their last nine league games was just about an upturn in morale.
"It's always tough to see," he added. "The manager takes the brunt and that's unfortunate because you can't get rid of a full squad of players.
"Craig has been here a long, long time ... he hasn't really changed a lot, it's been about getting the spirits up and getting people firing again."
Leicester have an opportunity to move into the top half of the league when they host 13th-placed Watford on Saturday.
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