Usually, I think,
your Tav gets a key from Entharl after she and Entharl speak.
If the chat is bugged, you might try just walking up to the door that leads to the blacked-out room and opening it, because maybe the key got dropped into your keys pouch anyway.

Barring that, I think picking the lock is your best bet. Going down the ladder and starting a conversation there should proceed the quest.

"Often forcing his victims to eat their own lips, he was caught and imprisoned for tax evasion." -Yellowbeard.