Why didn't we find katas for rationality?

The main difference between Katas and Drilling (at least most of the time) is that Katas are usually long chains of moves, and drills are a repetitive practice of a single, short move.

Another difference is drills are more often practiced with another person rather than on the air (even hitting a bag is still better than hitting air).

Krav Maga style defenses are in the middle, they don't have you do lots of moves out chained together of context in the air, but are also not just one move, they're usually ~5 moves intended for a specific situation practiced on another person with slight resistance. 

Well, they're mastery of a different thing. They're great for entertainment and cool looking demonstrations.

This is very true, but I think it misses a key point in what makes katas useful for actually learning a martial art in the first place. As noted in Matt Goldenberg's answer, partner work is much more important for actually learning to use a martial art. Just practicing a kata by rote may look pretty, but it won't tell you anything about how to use it. My own best teachers would teach moves and combinations and katas by rote at first, then very quickly move on to exercises that require creative application. Things like: 

• Mushin practice - get attacked and respond quickly, and make it work
• Randori practice - get attacked repeatedly by multiple opponents, and make it work
• Sparring and grappling practice

All of these can be modified with constraints to make them easier or harder. Easier might be "Every attack against you will be this kind of punch." Harder can be something like "Choose one part of one kata. Come up with a way to use it effectively against whatever your opponent decides to throw at you." Or "Figure out X different ways to use that same sequence of moves, with only minor variations, in different situations, then execute it and see how well it works." Hopefully you've been practicing smart all along, and visualizing opponents while practicing your katas in the air, and developing your understanding of the moves and body mechanics so your interpretations make sense!

To bring this back to the original question: what are we thinking of as the fundamental "moves" we're stringing together to make a "kata"? And is a rationality "kata" practice a one-person or two-person activity? Maybe a rationality kata is made up of moves like "1) Figure out what question/problem you're trying to solve. 2) Spend five minutes thinking about what you know that may be at all relevant. 3) Make a list of unknowns that would be useful to know, and estimate how hard it would be to get the answers. 4) Make a list of strategies and techniques you might try to find a solution, and estimate their odds of success and time required. 5) Rank the results of #4 however you like, and work your way down the list until the problem is solved. 6) Go back to #2 and repeat until you have a good enough solution." Or they can be more specific - katas for calibration, katas for probability estimation, and so on. "Practicing" katas, then, would be a matter of repeating the words of the steps, to commit them to the mental equivalent of muscle memory. This will be partly crystallizing useful concepts into short handles that leap to mind when needed, a kind of mental Miyagi-ing. 

Also, in martial arts, katas can be practiced many different ways. For my own black belt test, some of the kata portion included things like "Precisely execute every kata you know, in order, in less than X minutes total," and "choose one kata you know, and execute is as slowly as possible, minimum Y minutes, while maintaining precision and focus on every part of every movement," and "perform this kata correctly, but make it fit completely in a box no more than Z feet across." I think I've also been asked to do the mirror image of a kata a few times, though not as part of a rank test. Similarly, analyzing and verbalizing why a kata is a certain way, and what it's trying to explore and teach, can be helpful for guiding further practice.

Then, more applied practice might look more like randori, where you get random problems thrown at you repeatedly and you have to use a specific kata to solve them. Or like mushin, where you have to practice quickly figuring out which kata to use for a problem that gets thrown at you, and then make that one work. 

And yes, I realize I've just accidentally come very close to describing some of Jeffreysai's [LW · GW]methods, which is obviously not a coincidence, I'm sure.

Same as posts on fermi estimates. I just make a guess to whatever level of effort seems appropriate for the query (often pretty casual, but will take a bit more time if it's about something I feel is important or especially uncertain about). Then when I am getting the piece of info I can reflect on reasons I might have been off. This often helps structure my inquiry into the piece of info also as I map out model differences that are why it surprised me.

Varying the problem helps, as does varying your approach to the problem.  Studying math generally involves many years of working progressively complex problems.  But this is different from a "kata", which is a set of moves rigorously repeated in a specific order and invariant manner*.

Psychologically speaking, a kata functions to take a set of moves that the student consciously understands and build muscle memories that can execute the moves effectively at the sub-second timescale of a fight. Reasoning uses different cognitive systems, although once a behavior is consciously understood it is often useful to practice until mental subroutines are developed that enable quick, unreflective execution of the behavior.

*There will be some variation as the student goes from not being good at the kata to being very good at the kata, but that variation is unwanted.  The better you are at the kata the less variation there is.

It's also helpful to practice when you're first learning something.

I can think of a few skills that, while not "rationality" in themselves, make it much easier to reason effectively.  Numeracy is one.  The innumerate can't really see the difference between a million, a billion, a trillion, and a godzillion.  

It helps to have, in memory, a set of references to compare to.  For example, there are about a third of a billion people in the United States.  Therefore a billion dollars is roughly $3 each, a trillion dollars is roughly $3,000 each, and a million dollars is roughly nothing (.3 cents) each.

A working knowledge of history is also helpful, as is a rough understanding of manufacturing.

Perhaps, but it would surprise me if you don't have hundreds of common sudoku patterns in your memory.  Not entire puzzles, but heuristics for solving limited parts of the puzzle.  That's how humans learn.  We do pattern recognition whenever possible and fall back on reason when we're stumped.  "Learning" substantially consists of developing the heuristics that allow you to perform without reason (which is slow and error-prone).

In my opinion, what we focus on is sort of like SCUBA diving. Some overall training is helpful, so that you can understand what the equipment is for and how to use it. It does decompose into parts. But it's not a high-pressure sport, and skill comes from mainly going through the overall process and dealing with the random challenges that come up in individual circumstances. So you could teach it as "katas," but that would be a relatively inefficient way to learn.