Note to ESL learners: This sentence can be read as "That 'Ayumi' kid, [she is] only 13 but...."

Shouldn't there be another comma after "only 13"?

In my humble opinion, you don't necessarily need a comma in this context, before "but".
I also like commas (and punctuation marks in general, just as you do), but I didn't see any comma missing before you pointed out the fact.

I've just made a small experiment. I fed this short text into a program that checks texts to verify correct vocabulary and punctuation marks; the text received a score of 100 both without the comma and with it. ☺

I see. Okay. :)

Instead of a comma, the word "is" would be appropriate to match the Japanese.

That 'Ayumi' kid is only 13 but ...

I wonder about the ' ' around Ayumi, though.

"That Ayumi kid" seems like a bit too literal a translation to me.
In real life, perhaps we'd just say "Ayumi" in the same situation a Japanese would use that phrase.

Vortarulo,

You were probably thinking of a sentence structure like this.
In this case the "only 13" would be set off by commas.

That 'Ayumi' kid, only 13, has performed in movies and theatres as a child actor since she was five.

Yes, I think so. I still can't quite parse the sentence. I like the suggestion of adding an "is" instead of the comma, otherwise the main clause doesn't seem to contain a verb.
But I'm no native speaker of English, so you decide. :)

Also, perhaps the '...' around her name aren't necessary. It looks like it's only a nickname.

I'm also in favor of adding "is" instead of the comma.