Is it permissible for her to give money with the intention of zakaah on her own behalf and ongoing charity (sadaqah jaariyah) on behalf of her father?
It is permissible to give charity on behalf of another person, whether it is ongoing charity or a one-time payment. Ongoing charity is a waqf, i.e., the basic wealth or property remains and its yield is used to benefit others, such as if a person gives a house as a waqf for poor people to live in or to benefit from its rent, or he donates Mushafs and books of Islamic knowledge, or he builds a mosque, or he digs a well from which people drink. This is what is meant by ongoing charity.
With regard to giving food or cash, this is a one-time act of charity and is not sadaqah jaariyah (ongoing charity), but both types are good.
Muslim (1630) narrated from Abu Hurayrah (may Allah be pleased with him) that a man said to the Prophet (blessings and peace of Allah be upon him): My father died and he left behind some wealth but he did not make a will. Will it expiate him if charity is given on his behalf? He said: “Yes.”
Muslim also narrated (1004) from ‘Aa’ishah (may Allah be pleased with her) that a man said to the Prophet (blessings and peace of Allah be upon him): my mother died suddenly and did not leave a will. I think that if she had been able to speak, she would have given charity. Will she have any reward if I give charity on her behalf? He said: “Yes.”
An-Nawawi (may Allah have mercy on him) said: This hadeeth indicates that it is permissible to give charity on behalf of the deceased and it is mustahabb to do so, and the reward thereof will reach him and benefit him, and will also benefit the giver. All of this is agreed upon by the consensus of the Muslims. End quote.
See also the answer to question no. 122361
Based on that, there is nothing wrong with you giving some of your wealth in charity on behalf of your father, whether it is ongoing charity or a one-time gift. But that should not be from your zakaah, and it is not valid to combine the two, because zakaah is an obligation that must be fulfilled; it is an obligation upon the individual and it is not valid for it to be on his behalf and on behalf of someone else. Charity, on the other hand, is voluntary and it is permissible for a person to make its reward shared between him and someone else, such as if he has half of the reward and another person has half, as was stated by Imam Ahmad (may Allah have mercy on him). Ibn al-Qayyim (may Allah have mercy on him) said: What this means is that the reward belongs to him and he may give all of it or some of it. If he gives it to four people for example, each of them will have one quarter of it. If he gives one quarter and keeps the rest for himself, that is permissible just as if he gave it to someone else.
End quote from ar-Rooh, p. 132
To sum up: it is obligatory for you to give zakaah on your wealth, and to give it to those who are entitled to it, with the intention of giving zakaah, not the intention of giving charity on your behalf or on behalf of your father.
Then it is mustahabb for you to give something of your wealth in charity on behalf of your father.
Your giving zakaah to those to whom your father used to give zakaah is something good and praiseworthy, because it is upholding ties with those with whom your father used to uphold ties, so that they will not feel that they have lost him and they will continue to offer supplication (du‘aa’) for him and for you. The same also applies if you give charity to them.
We ask Allah to help you and enable you to do righteous and good deeds.