After flawlessly doing diplomatic immunity, I leave the Embassy and go to meet Delphine in River Wood; after retreaving my items from the chest I talk to her. She mentions something about Esbern and the 30th month after it registering the quest complete and giving me the achievement, I'm left alone with no main Quest. Anyway to get A Cornered Rat quest?

What does it say in your journal? Go to Riften and start it yourself.

That doesn't work. It's a broken quest line. Best bet is to re-load a save prior to turning in the quest and hope Bethesda patches it.