Praise be to Allah.
What is required of the healthy person is also required of the sick person in terms of purifying himself with water in the case of both minor and major impurity. So he should do wudoo’ in the case of minor impurity and do ghusl in the case of major impurity. If he is not able to do wudoo’ for himself, then someone else may help him to do wudoo’.
Al-Bahooti (may Allah have mercy on him) said in Kashshaaf al-Qinaa‘ (1/103): If the individual who is missing an arm, and the like – such as cases of paralysis, or a sick person who is not able to do wudoo’ himself – has someone who can help him to do wudoo’, or to do ghusl, in return for an appropriate fee that he can afford to pay, then he must do that, because he is subject to the same ruling as one who is healthy. End quote.
If the sick person is not able to do wudoo’ himself, and there is no one who can help him to do wudoo’, then he must do tayammum, even by placing his hands on the wall or on the bed, if there is dust on it, or he can keep some earth with him in a vessel or a bag with which to do tayammum. If he is not able to do tayammum either, then he must pray as he is.
Al-Bahooti (may Allah have mercy on him) said in Kashshaaf al-Qinaa‘ (1/103): If there is no one who can help him to do wudoo’ or tayammum, because he cannot afford to pay someone to help him, or he is not able to find someone whom he can hire to do that, then he must pray as he is. End quote.
Based on that, it is permissible for you to help this sick person to do wudoo’, by pouring water on him, and he can wash his limbs, or you can wash his limbs if he is not able to do that himself.
With regard to rinsing the mouth and nose, if the sick person is not able to do that himself, and it is too difficult for him, then there is nothing wrong with omitting it, but after finishing wudoo’, he should do tayammum, because he is regarded as having omitted washing one of the parts that must be washed in wudoo’. In the answer to question no. 88066, we have explained that wudoo’ and ghusl are not valid without rinsing of the mouth and nose.
With regard to tayammum, if he is not able to do tayammum himself, then you should place your hands on the dust, then wipe his face and hands with your hands.
And Allah knows best.