As written, but it’s dumb—the empowering is permanent
RAW, you can enter Deadhand Style, and then spend a ki point to empower your unarmed strikes to cause the shaken condition, and that’s it, end of the rules. Nothing there ever ends the empowering of your unarmed strikes—so it’s permanent. Any effect using ki is presumably supernatural, which means this empowering can be suppressed by antimagic field, but otherwise, it’s not going anywhere. Even disjunction does nothing to do it (probably; see this answer for a discussion of disjoining supernatural effects—Pathfinder hasn’t added any clarification on this as far as I know).
I think it’s pretty clear that there would be no point in having a style that you only ever needed to enter once, nor in a ki cost that would be refunded in a day and never paid again. But this is all we have, as written.
As intended—completely unknown
I can find no outside discussion of Deadhand Style by anyone from Paizo. There is a paizo.com thread asking both of your excellent questions, but it only got a few replies, none from Paizo staff, and none of which had an answer to this question (and whose answer to the other question was, officially, wrong, in the way I described in my own answer to that question).
As I interpret—sadly, probably just one attack
Because the feat doesn’t specify about self-stacking, which is usually a feature of fear sources that cost something to re-apply, it seems to me that you’re meant to spend a ki point for every application of this effect, so you spend more ki to accumulate the feat in the target.
This sucks because of the swift-action requirement—that means you can only pump the fear status once per round, and that means you take 3 rounds to reach panicked. After 3 rounds, a Pathfinder combat is almost-certainly over—a typical “striker” expectation is to finish a target in 1 round of focused attack (usually a full-attack). So much happens in a Pathfinder round, between extremely powerful spells, very-high damage from full-attacks, and so on, that it’s unlikely for there to be very much fight left after everyone’s gone once or twice. There may still be combatants standing but in most cases, one side or the other has gained a decisive advantage and now it’s a question of retreat versus finishing the enemy off. So by the time you manage to end one target’s threat via fear, another character may well have ended the threat of multiple targets, quite possibly the entire encounter.
Still, Deadhand Master is a pretty strong feat. Deadhand Style and Deadhand Initiate are pretty meh, but an actually-decent feat for a monk is a rare thing—two feat taxes is bad but not exactly out of line, considering the available precedents. Plenty of feats that require two bad feats are, themselves, bad, so this is still ahead of the game, sad as that is.
As I rule—one round
Personally, I’d rule this as having the empowering last one round. That potentially could allow you to bring someone all the way to panicked for just one ki point, but you have to hit at least three times, and then the target has to fail three Will saves—failing one Will save is usually enough to take someone out of a fight, so this is still pretty weak, but at least it’s worth using.