I had this problem myself. It's listed on the UESP page: http://www.uesp.net/wiki/Skyrim:A_Cornered_Rat
The quest may also fail to start after completing http://www.uesp.net/wiki/Skyrim:Diplomatic_Immunity, and even trying manually start or end it with the console will not work. You may have to skip forward to the start of the next quest, http://www.uesp.net/wiki/Skyrim:Alduin%27s_Wall, by using the console and typing in 'setstage MQ203 5', which will allow you to continue.
That's somewhat wrong, though. It doesn't skip to the start of the next quest, but rather to a point roughly midway along it. It actually teleports you, Delphine, and Esbern to the spot you need to be. Also, Esbern won't give you any dialogue from that point forward; he just keeps saying 'go away! I'm not opening the door!' Imperfect solution, but it will allow you to continue with the main quest.