Seeing as you are on the Dark Drinker quest. You need to kill the Void Knight (purple shield). Once you killed it you get a 30 seconds buff. (Warning: This buff does not show up on screen, but it does show up in you character screen, same place where buffs like Three of Coins show up.) You have to kill Alak'hul while the buff is still active. (You cannot kill your Knight after Alak'hul, as you need the buff from the Knight for it to work.) Best way to do it is to get Alak'hul really low on health, (you can kill the Solar and Arc Knight, you don't need those,) then kill the Void Knight and then burn down Alak'hul. (Don't be too fast though, if you kill them too quickly after eachother it might not count, 30 seconds is plenty of time, enough time to go into your character menu to see if the buff is active.) You got this. 👍🏻 [spoiler]People may not realise it yet but, these are Destiny 2 Forums now, there's a seperate section for all Destiny 1 related things all the way down at the bottom. 🤷🏻‍♀️[/spoiler]

Re-read the quest. You have to kill the corresponding element knight, then within 30 seconds of his death, finish off Alak-Hul.

You need to kill the corresponding Knight + Alak-Hul within 30 seconds of each other.