Seriously. How early is it to teach them about 4 of a kind and straights? I remember my dad teaching me while we were on an airplane, but I don’t remember when.
The kid in question doesn’t seem to like to be coached. She’s nine, and she’s done some individual sports, but she wants to do things her way. That’s fine, to a point.
Nine is easily old enough. She won’t get the strategy and will need help counting points for a while, but the basic rules are easily within her grasp.
Having her sit and be taught those rules without one or both of you getting so frustrated you quit, is another matter entirely.
Lotsa luck.
Sneak it in when they’re not looking. 
When they aren’t sitting being expected to listen, when engaged in doing their own thing, situations create opportunities, keep an eye out and take advantage of the opportunity as they come fleeting by. {well mind you I’m dealing with a 14 month old right now so this a bit skewed, excuse me, still that curiosity doesn’t die when it’s nurtured and as the kid grows older. With the five and eight year olds next door at home, same story different season. All they need is loving nurturing.}
Well it helps to set up the environment for those moments to unfold.
PS.
I’m also pretty sure Lausten has it figured out and the kid is in good hands. And that she’s going to give him a run for his money before he knows it. Guess it comes down to how much she wants to learn it.
both of you getting so frustrated you quitI did that years ago when trying to show an even younger kid how to throw a frisbee. That went south real fast. It's easier when their younger though, just remember everything is new and fun, so set it up so they can do something, and drop it as soon as they are not having fun, something else will replace it. But, as puberty begins, and they get beyond "because I said so" to having reasons for things, now, everything is a battle of wits.
I always keep in mind something a parent of teens once told me. At some point, your only job is to keep them killing themselves until they reach 18.
My friends are avid board gamers, favoring complicated games that take hours to play and have massive rule books (Agricola, Scythe, Twilight Imperium, etc.). They have a 12 and a 14-year-old who play many different games of that type at a high level (they regularly win in 6-person games).
I also have adult friends who we’ve tried to teach fairly simple board games to (Splendor, Century Spice Road, Kingdomino, etc.), and they never seem to get it. The rules and any kind of strategy are seemingly beyond their comprehension.
So know that different people have different abilities to learn certain kinds of things. And especially with kids, regardless of their ability to learn, if they don’t want to learn they ain’t gonna.
I’ve found the best way to teach a kid to play something is to play it while they watch over your shoulder so you can explain what you’re doing and why. Once in a while ask them what you should do to get them thinking and involved. Absorbing knowledge while observing is how kids learn 99% of stuff, so they’ll naturally absorb both the rules and the reasons behind the actions. Just giving them a bunch of cards and rules to follow is too much like school and turns them off.
We tried playing a few rounds with the cards face up. She was bored but stuck with it. She didn’t like the word “suits”.
So we played Spoons.
“So we played Spoons.”
Baby steps.
You’ll get to Rummy and beyond eventually.