No official rule
There isn’t some rule you’re missing somewhere in Player’s Handbook II that answers this question, nor does the shaken condition (or fear generally) have a default duration or anything, and of course the Player’s Handbook II errata—which does exist—doesn’t correct this.
The sort of obvious answer is probably 5 + Charisma bonus rounds, matching the duration of fighting challenge and test of mettle. The only knight’s challenges that don’t use this duration are call to battle and bond of loyalty, which are instantaneous effects and so don’t have a duration, and loyal beyond death, which is the 20th-level capstone ability and very special (burn an additional use of knight’s challenge for each round you want to keep it going).
The other option is to see daunting challenge as a kind of fancy demoralize (as in the Intimidate skill usage that also causes the shaken condition), and copy the 1-round duration of that. After all, daunting challenge affects multiple enemies, where demoralize only affects one, by default, and daunting challenge is a swift action instead of a standard. There are a few problems here, though:
- daunting challenge is a special, high-level class feature (anyone and everyone can demoralize, right from 1st level)
- it requires a limited daily resource (skills can be used as much as you want)
- and most notably, daunting challenge is limited to only affecting enemies considerably weaker than you (demoralize can target anyone if your Intimidate check is good enough)
So while daunting challenge does have a potentially-large advantage in terms of how many targets it can affect, and a definitely-significant advantage in terms of actions used, it’s got a ton of disadvantages that also having a decent duration would be appropriate. Also more broadly, demoralize has a lot of options for improvement, including ways to hit multiple targets, do it faster, or having a longer-lasting shaken effect. For that matter, the Intimidate check for demoralize can be made far more reliable than getting enemies to fail to save against daunting challenge’s DC. Daunting challenge is pretty much just stand-alone; the only ways to “optimize” it are to take more knight levels and increase your Charisma, to improve the saving throw DC (and get more knight’s challenge uses).
The third option is to do something in between. Maybe not 5 + Cha, but some other number. Whatever. Ultimately, it kind of doesn’t matter that much: a round is a long, long time in D&D 3.5e, and as soon as you make the duration more than one of them, it’s probably “enough.” A fight probably isn’t finished after 2 rounds, but there’s a decent chance it’s decided after 2 rounds—and as you get higher, that becomes more and more certain.
Really, the primary thing about daunting challenge that’s especially good is that it isn’t Intimidate—which means you can do both. Since fear stacks, that can be effective. But there are simply much easier ways to escalate demoralize than 12 levels of knight.
Also, as long as you’re aware of what you’re getting into, you do you, but just in case you aren’t aware: the knight is a deeply weak class. Bulwark of defense, test of mettle, and loyal beyond death are the only notably good class features it has—and you might recognize one of those as not being available until 20th. So I sort of suggest you don’t bother ever getting daunting challenge in the first place. It’s a powerful enough effect in theory, but the situation where it is powerful—large numbers of weak enemies—isn’t something adventurers run into very often, and when they do, isn’t usually much of a threat to them. If they’re protecting something much more vulnerable or fragile than themselves, it’s much better, but we’re getting into pretty specific situations that just don’t come up much at all in most adventurers’ careers. If knight was good, it’d be a nice extra. But knight’s not, and getting it costs, approximately, 8 levels of “not much.” That’s absurdly expensive.
