The Republic will finish in a disappointing fourth place in Group C, barring a cricket-score victory over the Kazakhs and an unlikely defeat for Austria in the Faroe Islands on Tuesday night, when second behind winners Germany was always their target.
However, the performance the interim manager got from his players in their 3-0 defeat by the Germans in Cologne on Friday night provided him with encouragement and he senses there is more to come.
Asked how important it was to end with a win, King said: "It would be lovely. But these players are up, they know they have acquitted themselves well against the top, they know that and they just have to repeat that.
"They are up for it. I know by their eyes, I know by their antics, I know by the way they are going around the place - they want to win the game.
"If you start a game in that mode, well that's all you can do and then you have to wait and see how that unfolds."
Victory would represent a personal achievement too for the Republic Under-21s manager, who was drafted into the hot-seat in the wake of Giovanni Trapattoni's departure last month.
King has been told he will not be considered for the job on a permanent basis, so Tuesday's game could be his final opportunity to lead his country to a win.
He said: "It would be lovely, wouldn't it? It would be lovely.
"I am happy to go to Germany - I thought that was a great football experience. The Cologne Stadium is marvellous, the players we played against are marvellous.
"That was a football occasion. We acquitted ourselves very well and on another day we may have scored a couple of goals, and on another day the deflected goal may not have gone in.
"But that's football and we live to fight another day. It would be marvellous to win. If we don't win, I have no issue on results.
"What I am looking for is can we dominate the game? Can we create chances? Can we score chances? Can we ensure clean sheets? These are the facts of football, and you get on with that.
"If it works - sometimes it does and it's magnificent; other times it doesn't and it's horrendous. That's it."
Captain Robbie Keane handed King a major boost by resuming training earlier than expected after sitting out Friday's game with an ankle injury. He had been a big doubt, but played a full part in Sunday afternoon's session and could yet be fit for Tuesday.
Defenders Richard Dunne and John O'Shea have also returned to the squad following suspension.