You must have traveled with Aviendha for 2 days.
You must have killed the first dragon, and become Thane of Whiterun(which happens when you complete the Dragon Rising quest).

Starts with Aviendha asking to talk to you, or you may enter dialogue yourself.

Must have completed quest 1.
5 days must have passed since quest 1 was completed.
Will only start in a city or town.

An event will take place in a random place.

Shout at it.

I opened the magic door and killed the enemy inside. Nothing more is happening.

You should find Aviendha shackled behind a magic wall. If she's not there(usually because she was managed with some follower framework), try the following.

Open the console and type: 
getstage _wotkidnap

Is the stage at 52? Then type
setstage _wotkidnap 55
Approach the wall and Aviendha should be there.

If it's already at stage 55, try setstage _wotkidnap 60
Other quest stages to try:
setstage _wotkidnap 70
Then talk to Moiraine.

Remaining quest stages:
80
85
90
100
120

Is _wotkidnap not found? Replace _wotkidnap with the ID using a method below.

Type in console: help Darkfriends
WoT.esp QUST: _WoTKidnap (073CCF57) 'Darkfriends'

Replace _wotkidnap with 073CCF57, i.e getstage 073CCF57

OR

Type in console: save funclist 1
Notepad will open, search for Darkfriends. A line like below will show.
393CCF57    13/13    00000002  78 Darkfriends                             - Quest Flags(2)     

Replace _wotkidnap with 393CCF57, i.e getstage 393CCF57

Must have completed quest 2.
1 day must have passed since quest 2 was completed.

Starts when you read a courier note.

Completed quests 3.
5 days must have passed since completing quest 3.

Starts when you read a courier note.

Completed quests 4.

Talk to Leane in the White Tower. You can travel to the White Tower via the portal stone in Mzulchond(near Markarth).

Completed quests 5.

Talk to Egwene who will approach you when you're in or near a major city, after completing The Forsaken.

Completed quests 6.

Starts when you return to the White Tower after completing Callandor.

Completed quest 7.

Starts when you read a courier note. It's delivered 2 days after the Broken Tower ends.

Completed quest 4.

Starts with Aviendha asking to talk to you, or you can enter dialogue yourself.