Does a foul odor emanating from the anus due to an illness, without the sensation of gas being expelled, nullify wudu, and is it permissible to repeat the prayer based on the opinion of Shaykh al-Islam who states that repetition is not obligatory?

If you confirm the presence of an odor before, during, or after ablution, or during prayer, your prayer is invalidated and must be repeated, based on the Prophet, peace and blessings be upon him, who said: "One should not leave [the prayer] until he hears a sound or perceives an odor." You must make up all the prayers you performed with that invalid ablution. If you know their number, make them up. If you do not know the number, continue making them up until you are reasonably certain that you have fulfilled your obligation. This is the view of the majority of scholars. There is another opinion held by some scholars, such as Ibn Taymiyyah, that repetition is not obligatory for someone who unknowingly omitted a condition or pillar, and a layperson is permitted to adopt this opinion if they are unable to delve into the sayings of scholars, provided that they do not intend to selectively follow lenient opinions.