Praise be to Allah.
It is obligatory for the children to help their mother to purify herself, because this is her right and is a duty upon them. Not doing that means that they are committing one of the most grievous of major sins, which is disobedience towards mthers.
There is the fear that Allah may hasten the punishment for them because of this sin.
So they have to repent to Allah, regret what they have done, apologise to their mother, ask her to forgive them, and strive hard to obey her and please her, for this is one of the gates of paradise and they do not know when it will be closed for them, at which time regret will be to no avail.
With regard to purifying herself if they refuse to help her:
If a sick person is unable to do wudoo’ and cannot find anyone to help him do it, even in return for payment, then he should do tayammum. He may do tayammum using the bed on which he sleeps if there is dust on it, or he can keep a vessel next to him in which there is sand or dust with which he may do tayammum.
Al-Bahooti (may Allah have mercy on him) said in Kashshaaf al-Qinaa‘ (1/103): If the sick person who is unable to do wudoo’ for himself can find someone to do wudoo’ for him or to do ghusl for him in return for an appropriate payment and he can afford it, then he must do that, because in that case doing wudoo’ is obligatory for him as it is for one who is healthy. If he cannot find someone to do wudoo’ or tayammum for him, because he cannot afford to pay or he cannot find someone to hire for that purpose, then he should pray as he is. End quote.
If the woman asked about here is able to hire someone to help her do wudoo’, then she must do that.
If she cannot afford it, then she should move on to doing tayammum, because Allah, may He be exalted, says (interpretation of the meaning): “But if you are ill or on a journey or any of you comes from answering the call of nature, or you have been in contact with women (i.e. sexual intercourse) and you find no water, then perform Tayammum with clean earth and rub therewith your faces and hands” [al-Maa’idah 5:6].
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: If a sick person does not have any dust, can he do tayammum using the wall or bed, or not?
He replied: If the wall is made of “clean earth”, whether it is rocks or bricks, then it is permissible to use it for tayammum. But if the wall is covered with wood or is painted, if there is dust on it, then it may be used for tayammum and there is nothing wrong with it, and he will be like one who does tayammum using earth. But if there is no dust on it, then it does not come under the heading of “earth” and cannot be used for tayammum.
With regard to the bed, we say: if there is dust on it, it may be used for tayammum; otherwise it may not because it does not come under the heading of “earth”.
End quote from Fataawa at-Tahaarah, p. 240.
If she is unable to do tayammum, she may pray without wudoo’ or tayammum, because Allah, may He be exalted, says (interpretation of the meaning):
“Allaah burdens not a person beyond his scope”
“So keep your duty to Allaah and fear Him as much as you can”
Thus it is known that it is not permissible for her to pray without wudoo’ or tayammum so long as she is able to do tayammum.
And Allah knows best.