Is it permissible to wipe over the socks for a long time because of illness?

The saheeh Sunnah indicates that wiping over the socks is limited to a specific time period, which is one day and one night for the non-traveller, and three days and nights for the traveller. 

Muslim (276) narrated that Shurayh ibn Haani’ said: I came to ‘Aa’ishah and asked her about wiping over the khuffayn. She said: You should go to (‘Ali) ibn Abi Taalib and ask him, for he used to travel with the Messenger of Allaah (S). So we asked him and he said: The Messenger of Allaah (S) set a limit of three days and nights for the traveller, and one day and night for one who is not travelling. 

Al-Tirmidhi (95), Abu Dawood (157) and Ibn Majaah (553) narrated from Khuzaymah ibn Thaabit (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) was asked about wiping over the khuffayn and he said: “For the traveller it is three (days) and for the non-traveller it is one day.” Classed as saheeh by al-Albaani in Saheeh al-Tirmidhi. 

The most correct scholarly view is that the period for wiping over the socks begins from the first time one wipes over them after breaking wudoo’, not from the time of putting them on. So if a person does wudoo’ for Fajr prayer, and puts on socks, then he breaks his wudoo’ at nine a.m. and does not do wudoo’, then he does wudoo’ at twelve o’clock, then the period begins from twelve, and lasts for one day and one night, i.e., twenty-four hours. 

Al-Nawawi (may Allaah have mercy on him) said: al-Awzaa’i and Abu Thawr said: The period begins from when he wipes over his socks after breaking wudoo’. This was narrated from Ahmad and Dawood, and it is the favoured view and the most likely to be correct based on evidence. It was the view favoured by Ibn al-Mundhir, and something similar was narrated from ‘Umar ibn al-Khattaab (may Allaah be pleased with him). End quote from al-Majmoo’ (1/512). 

Based on that, you should follow the period for wiping over the socks, and when it ends it is not permissible to wipe over the socks, until you take them off then put them on after purifying yourself fully. But the end of the period does not mean that wudoo’ become invalid, according to the more correct view, so if the period ends and you are still in a state of purity, it is permissible for you to pray with this purity until it is broken. 

What you mention about it being difficult to wash the feet can be overcome by sitting on a chair and the like, or pouring water over the feet without bending down to wash them. 

Al-Nawawi (may Allaah have mercy on him) said: Our view is that rubbing the limbs when doing ghusl and wudoo’ is Sunnah but is not obligatory. If a person pours water over it and does not wipe it with his hands, or immerses himself in a lot of water, that is sufficient for his wudoo’ or ghusl. This is the view of all the scholars except for Maalik and al-Muzani, who stipulated that in order for ghusl and wudoo’ to be valid. End quote from al-Majmoo’ (2/214). 

See also the answer to question no. 90218. 

We ask Allaah to heal you and make you well. 

And Allaah knows best.