I think that tenouchi—along with footwork—is one of the biggest "walls" of kendo. By that, I mean that if you don't figure it out and put the time in, your kendo level hits a wall. You just can't beg better at kendo, or at least your progress slows to a crawl. Tenouchi is easy to avoid doing because it's boring and hard. But if you put the time and work in, your kendo will improve and it'll just be more fun.
In our Zoom kendo keiko, especially, we do a lot of tenouchi drills. Due to how small and fast twitch the muscles and ligaments in your wrist and hand are, you shouldn't do a lot of tenouchi practice at once. Your muscles will recover quickly, so the best way to work on these is to do tenouchi exercise frequently but in short sessions. For example, doing 2-3 of the below drills, three times a day. Or using your grip exerciser every time you stop at a stoplight. In three months, you'll notice a big difference with your swing and shinai control, though the practice never ends and your kendo will only get better.
Lastly, before we get to all the drills, I want to note one thing: Pretty much every single one of these drills requires you to keep a strong grip in your bottom 2-3 fingers. That part of your grip should be tight, while the top two fingers should be loose, with space, allowing the shinai to move around. Once the grip of your bottom two fingers gets lose, stop, readjust, and start again. There's no point to doing these if your grip isn't tight. If you're having trouble doing any of these drills, I recommend starting with the hand grip (the last thing on the list). In the meantime, you can do these drills but I recommend only doing as many as you can do without your grip breaking.
In this drill, hold your shinai with one hand (right or left), place your hand in the middle of your body, above your forehead (almost like you're raising your hand up to swing). It's important you don't let your hand go off to the side, though. This will exercise slightly different muscles. From this position, don't move your arm, shoulder, or elbow at all. Only using your hand and wrist (we're trying to isolate and use only these muscles), rotate the shinai up and down, toward the sky. This drill uses gravity to add some weight to your tenouchi. I'd recommend 10-20 times per hand.
Extensions are similar to Upward Tenouchi, but they're a little closer to what tenouchi looks like in a real swing. We were doing these in our class, but Fumihide Itokazu gets credit for the idea of locking your elbow with your other hand (see video below). Hold the shinai in one hand, straighten that arm, and put the back of your hand on your elbow and pull back. Doing this prevents you from using your arm and elbow, which makes it easier to isolate your wrist and hand.
Tenouchi Extensions have a range that's way more than regular tenouchi in a swing, so go all the way back (almost touching your head) and extend all the way out (horizontal, or even a little lower). However far you extend, just make sure your grip doesn't loosen. If your tip is going way below horizontal, that's what's probably happening! I'd recommend 10-20 of these per hand.
I learned the upward version from Bryan Imanishi Sensei. I love this drill because it teaches you to keep the bottom two fingers tight while maintaining a space at the top of your hand. If you don't have that space, your whirlpool will be really, really skinny!
Put your hand in the middle of your body, a little ways out. With a good grip on your bottom two fingers, twirl your shinai tip around so that the top is wide and the bottom is as small as possible. Don't move your arm, elbow, shoulder, etc. Only your hand is moving, and (again) we're isolating those tenouchi muscles. I'd recommend 20 rotations in each direction, and both hands.
I was thinking the whirlpool drill is a good way to develop the control and strength you need to do maki waza and even harai waza. Here's another version with one hand in chuudan no kamae. It's the same thing at a different angle. Same thing, 20 times both directions, both hands.
Tenouchi Push-Ups don't help with your grip, but they do exercise the tiny, stabilizing muscles in your elbows and wrists. Have you ever swung hard and felt like the ligaments in your wrist were being pulled apart? That's what we're working on. Also the elbow portion of your swing.
To do these, just put your left hand in your right. At extension (pushed up position), your arms should be straight and perpendicular to your body. That's where your arms will be (ideally, at least) when you swing the shinai. This drill will help you to get that extension when you swing your shinai, too. Try to extend / push up sharply and snappily, because that's what you want with your swing. You can relax and fall back into the starting position a little more slowly if you want. Try to do 50-100 of these in one session. If that's too easy, you can make it harder by going faster, or even by putting more weight on one arm than the other (in theory, you could do these with one arm, too, but be sure to work up to that!).
These are similar to the Upward Tenouchi drill, but this time you're using both hands (choked down). I learned these from Fumihide Itokazu. I like it because you're using both hands, and there's way less chance for injury (compared to some of these other drills).
Like most of the other drills, we're trying to isolate the tenouchi muscles, so we want to limit how much we use our arms. Focus on your hands and wrists, bring the tip back without letting your grip loosen, then swing up to full extension. Like Upward Tenouchi, gravity acts as resistance to make these a little harder (and safer!). 20-30 times per session is pretty good for these. You can increase the speed and snappiness to make it harder, or switch to one hand (Upward Tenouchi).
In order to swing down, you have to swing up. This drill will help with small men or kote because you need to learn how to swing using only your hands and wrists (not pulling back with your arm).
From a one-handed chuudan no kamae, you're just "flicking" the shinai tip up without using anything else (no elbows, arms, or shoulders). Keep a tight grip on the bottom two fingers, and rotate your hand up. Your index finger will inevitably help, and that's okay. I'd recommend 10-20 times per hand.
I found this on YouTube somewhere, and thought it was pretty funny. But, it actually turned out to be a big part of my routine. It's actually really tough! Not only does it help you with grip, it works your core, too. It will help you to develop a fast, and strong, kote men. It's really the same motion but your hands are just in a different position.
To do Shinai Flails, find the center of gravity of your shinai. That's where it'll balance on one or two fingers.
Then, grab that spot with both hands, with the handle facing down. From this position, grip with your pink and ring finger on both hands. You're going to use your hands to rock the tip/handle back and forth, forward and back. Keep your arms extended, and speed things up. If you're hitting yourself in the face (or somewhere worse), you're loosening your grip too much.
I'm doing a bad job at this, but try to keep your core, butt muscles, and the rest of your body stationary as you do them. There's a lot of force in the flail, but the more solid your body is to go along with it, the stronger you're going to be able to hit using just tenouchi when you transfer this practice to your swings.
Another staple of Fumihide Itokazu, these are just great for beginners (and probably everybody else, too). They teach you how to use and move your hands and wrists quickly, just like you need to be able to do when you do tenouchi in your swing, big or small.
To do these, put your palms together and weave your pinky, ring, and middle fingers together, keeping your pointer pointed and extended. Put your hands in the middle of your body, next to your chest and voila! You're a ninja!
From here, just rotate your hands down. If you only use your hands and wrists, it should be easy and smooth!
Then repeat at a high speed!
One addition I thought about was doing progressive muscle relaxation in addition to these. To do that, you get into the starting position and then tense all your arm, shoulder, hand, etc., muscles as tight as you can. Then, after 5-10 seconds, relax and feel what it feels like to relax those muscles. The idea of progressive muscle relaxation is that it's much easier to feel your muscles engaged than disengaged. By engaging them, and then relaxing, you can feel how to disengage all those extraneous muscles so that all you're using is hands and wrists. This feeling will transfer over to your swing, too!
A Good Ol' Fashioned Grip Exerciser
Last but not least is a good old fashioned grip exerciser. You can probably grab one for $10 at the most. I'd recommend something that's pretty low resistance (that is, easier to squeeze), especially if you're just starting out. More squeezes and faster squeezes will get you further than something with a lot of tension. A hundred rapid fire squeezes in a row should take less than a minute. Do this 3-5 times a day. Figure out how to stack it on top of some other habit. Just do it, and do it often!
Pro tip: Flip your grip exerciser upside down to emphasize pinky/ring finger strength.
I hope these help you. Remember that hand strength and tenouchi take a long time before you notice your progress. What you're able to do now is just a lagging indicator of what you've been doing for the past few months. More than worrying about what tenouchi exercises to do, you should just figure out how to get yourself to do them regularly. What kind of triggers can you create to remind yourself? How can you shape your environment to make them easier to do?