From the Super Mario Wiki
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Like with other Donkey Kong boss levels, Donkey Kong has four lives and the amount of lives the player has is dependent on how many Mini-Marios were collected in the previous level. If the player selects this stage from the level select screen, they will begin with four lives.
There are several conveyor belts in the area that change direction whenever Donkey Kong presses the Direction Switch. Donkey Kong will throw pink Bob-ombs onto the conveyors and they will move toward the player. The player needs to pick up a Bob-omb and carry it up to the top conveyor so they can throw it at Donkey Kong. Every time Donkey Kong is damaged, he will throw Bob-ombs and press the Direction Switch more frequently, with him keeping four Bob-ombs on screen at all times when he has one life remaining. Attacking Donkey Kong four times will defeat him and the player will move on to Twilight City.