Saints had dropped to second after West Ham won on Saturday but victory would have seen them move back to the top of the table. However, they found themselves 2-0 down inside 26 minutes following goals by David Nugent and Matt Mills as they suffered their second successive home defeat and their third loss in four league games.
Adkins was realistic after the defeat, and told Saints Player: "We were up against it tonight. As I have said on many occasions, when things aren't going right that's when you find out about people and we have found out about all of us tonight in certain ways. Emotions are quite high at the moment but we will reflect on it in the cold light of day."
He added: "We need to work very hard to keep driving this club forward. We have to grind out performances and tonight wasn't good enough, we know that.
"We will reflect on the performance and reflect on the result, we still have a long way to go and we have to work very hard and stick together."
Leicester boss Nigel Pearson praised his side's work ethic as they moved to within five points of the play-offs.
Speaking at a post-match press conference reported on Leicester's official website, Pearson said: "I am very pleased for the players. They have put a lot in to the game tonight and they have got what they deserved.
"It is all well and good having talented players but whatever level you play at you must always work harder than the opposition and the work ethic really was exceptional.
"This is a long way to come for us all and for the supporters especially it is a great result.
"They made a tremendous noise and backed the players from the start right until the finish."