First, run "Tower Defence" mission from the tank near Susanoo's diner ASAP. That will get you second set of controllers so you can attack like normal against the aegis with both weapons. Setup your controllers with 3 you first got on one weapon and the 3 you got from Kurt on your other weapon.
Second, prioritize the main story missions from issue 9 as soon as you have both sets of controllers, that will get you 3 QL0 AE Capacitors. With just those, you will jump from doing 5% damage against aegis to like 15%. Capacitors are a much bigger boost to how much of your normal damage can affect Aegis shields compared to your controllers alone.
Third, prioritize attack rating talismans except for 1 or 2 health/healing talismans. It is going to take you a lot longer to burn AEGIS if majority of your talis are +HP.
While the pre-done build "Decks" ingame are usually rather lackluster, the Agent of Chaos (Elite) deck is very very good for Kaidan. Even ingame, it says it's designed to be used for solo play in Kaidan. It's got 2 really good impairs (The Art of War, Helter Skelter), 2 self-heal abilities (Turn the Tables, Martial Discipline (with the passive)), and a nice AoE builder (Escalation), AoE consumer (Clearing The Path), and single target consumer (Pulling The Strings).
Take a look at https://chronicle.thesecretworld.com/character/Kiraalyon if you want to compare. (Luck talisman doesn't show right for some reason, it's an issue tali from Issue 14, Black Pharoah Tarot with Ravaging Glyph and Signet of Set (+1.5% damage) on it).