This quest has a couple of nasty bugs. When you're told to break the gemstone to gain entrance to the next section sometimes the gem isn't there after you fight the monster guarding it. If it fails to appear you need to abandon the quest and restart. But, wait, that's not all -- restarting requires you to go back outside and talk to Queen Ayrenn and when you follow her in the first phase of the quest is missing and it puts you right back at the doors. Which still may or may not have the gems.
But even if you get lucky and manage to break your gem you still can't get thru the door until you figure out that clicking anywhere near the 'go thru this door' icon on the door won't work. You have to click on the far left or right edge away from the bluish glowing mist that's left behind because, I'm assuming, Ayrenn's gem is still there.
I can't believe this early quest in Auridon is so messed up. Please fix your game, ZOS.