Ki Strikes are part of the Attack Action.
Let me explain why.
This is the original text from the Core Rulebook, let me bold the important parts:
Ki Pool (Su): At 4th level, a monk gains a pool of ki points,
supernatural energy he can use to accomplish amazing feats. The number
of points in a monk's ki pool is equal to 1/2 his monk level + his
Wisdom modifier. As long as he has at least 1 point in his ki pool, he
can make a ki strike. At 4th level, ki strike allows his unarmed
attacks to be treated as magic weapons for the purpose of overcoming
damage reduction. At 7th level, his unarmed attacks are also treated
as cold iron and silver for the purpose of overcoming damage
reduction. At 10th level, his unarmed attacks are also treated as
lawful weapons for the purpose of overcoming damage reduction. At 16th
level, his unarmed attacks are treated as adamantine weapons for the
purpose of overcoming damage reduction and bypassing hardness.
There is no mention of what kind of action is a ki strike, so the rules say it should be a standard action, but since it does mention that his unarmed attacks are treated as , we conclude that Ki Strikes are part of the Attack Action. They dont take any kind of action, they are simply a special unarmed strike that has some ki, making it magical somehow.
I split the text in two parts so we can easily see the differences in the mechanics, notice bellow how three uses, all those that spend ki points, are said to use swift actions, and how they are on another paragraph.
By spending 1 point from his ki pool, a monk can make one additional
attack at his highest attack bonus when making a flurry of blows
attack. In addition, he can spend 1 point to increase his speed by 20
feet for 1 round. Finally, a monk can spend 1 point from his ki pool
to give himself a +4 dodge bonus to AC for 1 round. Each of these
powers is activated as a swift action. A monk gains additional powers
that consume points from his ki pool as he gains levels.
The ki pool is replenished each morning after 8 hours of rest or
meditation; these hours do not need to be consecutive.
We can sum up that the following abilities are Swift Actions and use up 1 Ki Point:
- Make one additional attack at his highest attack bonus when making a flurry of blows attack,
- or Increase his speed by 20 feet for 1 round,
- or Give himself a +4 dodge bonus to AC for 1 round.