Praise be to Allah.
We ask Allah to record for you the reward of paying attention to your sister and your keenness for her to pray regularly. And we ask Allah, may He be exalted, to guide your mother to undertake the duty of calling her daughter to do good, and to raise her properly. If the daughter refuses to pray, the mother should smack her (lightly) as a form of discipline, so as to save her from following her whims and desires, and so as to raise her to obey Allah, may He be exalted. And we ask Him to guide your sister and to make it easy for her to adhere to His commands.
The text of the hadeeth mentioned in the question is:
It was narrated that ‘Abd-Allah ibn ‘Amr ibn al-‘Aas (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “Instruct your children to pray when they are seven years old, and smack them (lightly if they do not pray) when they are ten, and separate them in their beds.”
Narrated by Abu Dawood (495); classed as saheeh by al-Albaani in Saheeh Abi Dawood.
The one to whom the command to teach the child is addressed is not only the father and is not only the mother either; rather it is addressed to everyone who is responsible for this child, but the father is the first one to be addressed, because he is the “shepherd” of his household, and he is the one who is responsible for his flock; the mother shares this responsibility with him.
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
It is obligatory for every person in authority to instruct those over whom he has authority to pray, even the children who have not yet reached puberty. The Prophet (blessings and peace of Allah be upon him) said: “Instruct your children to pray when they are seven years old, and smack them (lightly if they do not pray) when they are ten, and separate them in their beds.”
Whoever has a slave who is a minor, an orphan or a child and does not instruct him to pray, the adult may be given a severe disciplinary punishment (ta‘zeer) if he does not instruct the minor (to pray), because he has disobeyed Allah and His Messenger. The same applies to one who has adult slaves, servants, a wife or slave women: he has to instruct all of them to pray; if he does not do that, then he is disobeying Allah and His Messenger. End quote.
Majmoo‘ al-Fataawa, 22/50, 51
And he (may Allah have mercy on him) said:
Rather guardians have to instruct children to pray when they reach the age of seven, and smack them (lightly if they do not pray) when they are ten.
Majmoo‘ al-Fataawa, 28/360
Based on that, if your mother or father are present, they are responsible for teaching your sister and smacking her (lightly) if she has reached the age of ten (and is not praying). If your father is not present, and you have an older brother, then the responsibility is shifted to him and your mother.
For more information, please see the answer to question no. 126208
Your mother should also pay attention to herself, lest she bear the burden of sin for what she said. Allah, may He be glorified and exalted, will not call your sister to account for not praying if she has not reached the age of puberty, but the mother is accountable for not teaching her daughter and not encouraging her to pray, and not smacking her (lightly) if she does not pray.
Al-Maawardi (may Allah have mercy on him) said:
Teaching that to them before they reach puberty makes them get used to it and feel comfortable with it, but neglecting them and failing to teach them causes obvious harm, as it makes them lazy when it (prayer) becomes obligatory, and makes them feel reluctant to do it at the time when it becomes obligatory. As for teaching it to them when they are younger than seven, that is not obligatory; in most cases they do not learn properly what is taught to them and they are not able to do what they are told. But when they reach the age of seven, they are able to discern and learn properly what they are taught. At that age it becomes obligatory for their parents to teach them, but they do not have to smack them if they do not do it. When they reach the age of ten, it becomes obligatory to smack them if they do not pray, but the smack should not be painful or cause injury, and it should be in places where there is no fear of causing injury. And when they reach puberty, they are now regarded as accountable, so they should be reminded and they have to purify themselves, pray and do all other obligatory duties. End quote.
Al-Haawi al-Kabeer, 2/314
With regard to your role, it is to carry on advising and guiding your sister, and encouraging her to pray in a kind and gentle manner.
We ask Allah to guide her.
And Allah knows best.