I tried to piece the story together in my head from what I remembered.
Noble team and all of the troops were under Winter Contingency so that to the troops off of Reach the battle was only one day.
As said in the Journal, Halsey was at Sword base and began looking at classified files; thus learning of the Spartan III's and after learning more about them she learned they were a new series of Spartan and that Jorge was with them.
Then all the way to The package Halsey was keeping tabs on them (marking dead Spartans as MIA on her screen)
Now at this point it sort of blends into the books; Noble team heads off with the Cortana half and the Pillar of Autumn lands to recieve it. Cortana thus didn't know about the battle until they were in orbit and she was sort of too busy to mention it at that point. Also she must have understood she wasn't allowed to talk about it anyways.
As this was happening Jun escorted Halsey to Castle base. Once they arrived there a note I saw in the re-release of Fall of Reach confirms that Jun basically let the elites kill him so that Halsey could get away. she then retreated deeper into the base and as Castle base fell the Spartan IIs arrived and she ripped out the two pages in her Journal.
Halsey then eventually ended up on Onyx
That sort of works, generally, right?