For starters, this is a story with more than one reprint and with such things there is always the potential that a typo gets repeated reprint after reprint, but after some thought, I don't think that is the case. But before I begin, let me lay down what I think may be the exact DELE text:
Además, estaba cansado de las quejas de su hija porque sus compañeritos del banco de atrás, en la escuela, [les / la / le] ensuciaban las puntas de las trenzas en el tintero.
At the bottom of the page is this:
Adaptado de www.oni.escuelas.edu.ar. Argentina
I was unable to access the original from which this was adapted, so I could not see if there might be a typo or misinterpretation or omission of the original that makes this sentence difficult to parse/understand with certainty. But allow me to continue to answer this with a different perspective.
Initially, I thought this implied that his daughter and her classmates had complaints, but that is not what it says. It says, "las quejas de su hija" (his daughter's complaints). This passage is not discussing anyone's complaints but those of his daughter.
Before I go any further, let's analyze this sentence in English. Let's start with a translation:
Besides, he was tired of his daughter's complaints because her little classmates on the back bench at school were getting the ends of her braids dirty in the inkwell.
At first, I resisted this translation, but the more I thought about her little classmates on the back bench and the more I thought about the fact that the complaints weren't from all the little girls (just his daughter). The more I see the sense in this translation.
This sentence is a complex sentence — a sentence with a main clause and a subordinate clause. The main clause is:
He was tired of his daughter's complaints.
The subordinate clause, joined with because is:
Her little classmates on the back bench at school were getting the ends of her braids dirty in the inkwell.
In the subordinate clause above, we have a known actor — her little classmates (sus compañeritos). They are doing the action and the only action in the subordinate clause is getting the ends of her braids dirty in the inkwell.
What are her classmates dirtying? The ends of her braids. The ends of her braids is the direct object. The daughter is the indirect object. This construct is not dissimilar to any of the following:
Le dieron un bolígrafo.
They gave him/her/you a pen.
Le repararon el coche.
They repaired the car for him/her/you.
Le rascaron la espalda.
They scratched his/her/your back.
There's a lot of details in the sentence that throw you off, but basically this is just an example of a verb with both a direct and indirect object.
By the way, I did a search for "ensuciaban las puntas de las trenzas en el tintero" on Google. It looks as if most of the 48 pages with this phrase preceded "ensuciaban" with "le." It took me a while to understand why this is so, but now, like user 13628, I am most definitely on board.
Note: If I agree with another's answer on a thread, I usually don't bother to chime in unless I have something to add, but truth be told, I actually started this answer from a completely different angle and then changed it once or twice again as I researched it more closely. I think it adds to the conversation though, so, rather than delete it, I've decided to post it.
