The expiation for killing and the diyah is only required from someone who kills another person by mistake either directly or indirectly, as al-Hajaawi said in Zaad al-Mustaqni‘ (p. 222): Everyone who kills a person directly or indirectly must pay his diyah. End quote.
Killing directly means, for example, shooting at an animal (when hunting) and hitting a person instead, or rolling over another person in one’s sleep and killing him. Killing indirectly means, for example, digging a hole in the path used by people into which a person falls and dies.
With regard to killing directly, there is liability in all cases. With regard to killing indirectly, there is no liability or expiation required unless the one who did that overstepped the mark or was negligent.
It says in al-Mawsoo‘ah al-Fiqhiyyah (28/280): The general principle is that if the killing was done directly, there is liability even if it did not overstep the mark; if it was done indirectly, there is no liability unless he overstepped the mark.
In Qaraaraat Majma‘ al-Fiqh al-Islami (p. 81) it says: The one who does it directly is liable even if he did not over step the mark, and the one who did it indirectly is not liable, unless he overstepped the mark or was negligent.
The difference between overstepping the mark and negligence is that negligence means failing to do what is required of taking precautions, whereas overstepping the mark means doing actions or using things in a way that is not permissible.
End quote from al-Qawaa‘id wa’l-Usool al-Jaami‘ah, by Shaykh ‘Abd al-Rahmaan al-Sa‘di, p. 62
From what you said it does not seem that it was the result of any overstepping the mark or negligence on the part of the mother, because it is customary to leave children of that age without tying their wrappers.
Therefore the mother does not have to offer any expiation, because what happened was the will and decree of Allah and she had no hand in it.
We ask Allah to make this child a cause of her parents entering Paradise.
And Allah knows best.