This one can be kind of tricky since there are a number of events requiring you to clear out entire rooms to proceed, but it can easily be done in under 15 minutes on easy without glitching methods.
Room You Start In (~2 minutes in): Clear out the initial Covenant pack (2 grunts, 2 Brutes, +2 extra Grunts) Then clear out another wave that triggers near the exit (Brute, Grunts, Jackals). Then go back to the other end of the hall and clear out that pack (2 Brutes). And finally, kill the Brute that spawns on the ledge. After he's down Cortana will prompt you to plug her into a terminal and hurry to the exit.
Outdoor Terrace with Elevator (~3 minutes in): Another room that requires you to clear everyone to proceed. Don't bother, run past the enemies and jump down on either the left or right side of the far platform. Make sure to skid along one of the pillars on the way down to avoid a falling death and land at the bottom.
Room with Two Platforms (~3:45 in): Don't bother with killing anyone. Just run down to the first floor (you enter on the 2nd floor) and run into the bottom elevator. Wait a second for Cortana to trigger the lift and you should be down to the prisons.
Prisons ~7-8 minutes in): This requires you to do some clearing. Kill the initial guards on the top floor and Cortana will tell you where the first prisoners are. They alternate between the middle and bottom, so just hang out in the middle once you kill the initial guards. Then clear the guards for the prisoners to be released and do this twice. Once both groups of prisoners are free, kill the newly spawned enemies and work your way to the lift on the top floor. Here 3 Grunts, 2 Jackals and 2 Brutes will float in 1 at a time. Take them out, Cortana re-activates the lift, and go up.
After you complete the prison sequence, just run straight through the mission and don't bother killing anyone you don't have to. On easy, they aren't worth the trouble. But do watch out for charging Brutes who will knock you off the edge. And since you are running through the level, the last checkpoint will be back at the Prison, so DO NOT DIE HERE or else you will probably need to restart.
Room with Brutes vs Elites at End (~12-13 minutes in): Clear out every single enemy in the room as quickly as possible. Hunters will then spawn in as well as another group of Brutes. Kill them as well. Finally the door at the back of the room (the exit) will open and some elites with swords will rush through. Quickly kill them and run to the exit. At this point, you're home free to run to the exit.
Here's a speedrun that demonstrates these clear rooms. (He uses glitches, but really you don't need them)