Boy, kind of light on the spoilers in here.
You've seen Kirai's card, so I don't need to cover that.
Ikiryo is Kirai's own spirit. It's HER, just her spirit has manifested separately from her body. Can't buy it in your list, only summon it with the Summon Ikiryo spell. She's pretty nasty in melee but short on wounds. It also heals Kirai which is handy since she hurts herself a lot.
Lost Love is Kirai's totem, it's her dead boyfriend. He can heal Spirits and potentially grant Spirit to a non-Master model.
Datsue-Ba looks pretty unpleasant, and can mess with your opponent in a variety of ways as well as potentially summon more Spirits.
Gakis are cheap melee spirits, they can eat models like Hoarcats do and aren't too bad to summon. They also heal when they kill a model or eat a corpse.
Onryo are your main beatsticks I think. Good fighters, decent wounds, and a ranged attack too.
Seishin are cheap and drop easily but you will generate more so no big deal. They can buff other spirits but don't fight themselves.
Shikome are nasty geisha-harpies that run something down and kill it, think of the Prey ability in Warmachine.
I would advise a starting Kirai player to pick up the box, a blister of Gakis, and a blister of Seishin. Optionally, you may also want another blister of Gakis, Seishin, or Onryo. Datsue-Ba and Ikiryo are both unique so don't get extras of them. Best thing would be to play some sample games and see how many of each thing you need.
I have not gotten to play her yet, I'm mainly passing on the advice I got from the Henchman I spoke with at their booth.