Credit to TackiestTaco and Chaddiggans on Reddit for finding this out.

How to get the Black Spindle

Do the Lost to Light mission when it is the daily heroic story mission. Go through like normal until you reach the 3rd room you have to use relics to unlock the door. As you are facing the door, go towards the right and you’ll see a path leading to another door. Your ghost will tell you about a Fallen beacon, then you can enter the ketch. When you get inside, you have to kill all the Taken and a Taken Kell named Driviks, The Chosen. You only have 10 minutes to do this. If you do it in time, the mission ends and you get Black Spindle as one of the rewards.

Credit to TackiestTaco for taking the picture.

Some people have reported the door not being open when they get there, so there may be a yet unknown trigger - if you have ideas, leave them in the comments. In the video, they trigger the Black Spindle room as they are looking for the 2nd relic in the 3rd room.

I went in there with a friend and just passed everything up until reach the 3rd room. It was open as soon as we got there and we ran right in; it may be another time limit.

This is a hard mission so I suggest going in with a fireteam of 2 or 3 to complete it. Two of us went in and weren’t able to complete it in the 10 minute time, so I highly recommend 3. Focus the boss as much as you can, but don’t let the other enemies overrun you.

That’s all you need to get the Black Spindle in Destiny. Let me know if you have any questions or other strategies for finishing this mission.