He could not do wudu or tayammum in the hospital so he prayed as he was
Question

I went to hospital and stayed there for three days. I was not able to do wudu for prayer. I took a stone with me and did tayammum, but for Fajr prayer I could not do tayammum; I am paralyzed and I cannot move at all except with help. I did not want to miss Fajr prayer, so I imagined myself doing tayammum and prayed Fajr. Also, I was not facing the direction of the qiblah. Is what I did permissible or not? Do I have to make up the prayer?.
Praise be to Allah.If a person who is sick is not able to do wudu and has no one who can help him to do it, then he may do tayammum, even by wiping his hands on the wall or the bed, if there is dust on it, or he can keep with him some dust in a vessel or bag to use for tayammum. If he is not able to do tayammum, then he may pray as he is. 
The same applies with regard to facing the direction of the qiblah. Whoever is able to do that must do it, but if he is not able to do it, he may pray as he is, because Allah, may He be exalted, says (interpretation of the meaning): “Allah burdens not a person beyond his scope” [al-Baqarah 2:286]. 
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 
The Standing Committee for Issuing Fatwas were asked: I am bedridden and cannot move; how can I purify myself in order to offer the prayers, and how can I pray? 
They replied:  
Firstly: with regard to purification, the Muslim is required to purify himself with water. If he is unable to use it because of sickness or some other reason, then he should do tayammum using clean earth. If he is not able to do that, then the requirement of purification is waived and he may pray as he is. Allah, may He be exalted, says (interpretation of the meaning): “So keep your duty to Allaah and fear Him as much as you can”[al-Taghaabun 64:16] and “and [Allaah] has not laid upon you in religion any hardship”[al-Hajj 22:78]. With regard to what comes out, of urine and stools, it is sufficient to clean oneself with a stone or clean tissue, wiping the place three times or more until it is clean. 
Secondly: 
With regard to prayer, the one who is sick is required to pray standing; if he cannot do that then he may pray sitting; and if he cannot do that then he may pray on his side, because of the report narrated from ‘Imraan ibn Husayn, according to which the Prophet (blessings and peace of Allah be upon him) said: “Pray standing, and if you cannot then sitting, and if you cannot then on your side.” And Allah, may He be glorified and exalted, says (interpretation of the meaning): “So keep your duty to Allaah and fear Him as much as you can”[al-Taghaabun 64:16].
End quote from al-Fataawa al-Muta‘alliqah bi’t-Tibb wa Ahkaam al-Marda, p. 78 
If you prayed Fajr without wudu or tayammum, and without facing towards the qiblah, because you were not able to do that, or there was no one to help you do wudu or tayammum, or turn you towards the qiblah, then your prayer is valid and you do not have to repeat it. Similarly you do not have to imagine yourself doing tayammum, because this is not prescribed in Islam and is of no benefit. 
But if you were able to do tayammum or wudu, and face towards the qiblah, with the help of someone else, but you failed to ask for help, then you have fallen short in attaining purification which is a condition of prayer being valid, and you have to repeat the prayer that you offered without purification or facing towards the qiblah. 
And Allah knows best.