Most conditions, including Dazed, end whenever the effect that created them says they do.
If Player dazes Enemy until the end of Player's next turn, then the condition ends at the end of Player's next turn. It could also end if another, more specific effect ends it, like one of Enemy's or their allies' abilities. It would also end if Player delays their turn, as explicitly spelled out in the rules section concerning the Delay action (Rules Compendium, pg 242):
At the moment the creature delays, any effect that it has been sustaining ends. In addition, effects that last until the end of the creature’s turn now end if they are beneficial to it and its allies—they cannot be prolonged by delaying. For instance, if the creature stunned an enemy until the end of its next turn, the stunned condition ends as soon as the creature delays.
Note that this rule applies when the stunning creature delays, not when the stunned creature delays.
So, it goes like this:
- Player dazes Enemy until the end of Player's next turn.
- Everyone else goes in turn order, until Enemy's turn, right before Player's turn.
- Enemy's turn starts, but Enemy delays with a free action.
- Player takes their turn. If they delay, the daze effect on Enemy ends immediately. If they don't delay, the daze effect ends at the end of their turn.
- Enemy takes their turn. They will now be after player in the initiative order. The daze effect has ended, so they are not dazed and get all their actions.