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There is some confusion.
The quest “Into the Machine” is not an introductory quest, and it does not start in Dalaran.
So if that was the quest, you should be able to get back to Khaz Algar by going to the portal room in Stormwind and taking the portal to Dornogal.
The Portal to Dornegal is not in plain sight. When you run into the big cross-corridor where all the portal are, don’t turn - run straight across and around to keep going to the next cross-corridor.
Incidentally, as a Mage, a Portal trainer - certainly the one in Dornegal, but maybe the ones in Stormwind - should be able to teach you the spell “Teleport - Dornegal” so you don’t need portals in future.
If all this is wrong, and it actually was another quest you were on, please post again and we’ll try to figure it out.
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P.S.
I checked your quest history at
and you have indeed done all the quests up to “Into The Machine” in The Ringing Deeps, so that is indeed correct and where you are at.
Here is a YouTube that should show you the portal to get back to Dornegal.
Don’t forget to learn the spell Teleport: Dornegal
from Archmage Celindra
in the room where you arrive in Dornegal so you can zoom back there anytime without a portal.