What is the ruling on a spot that was splashed by impure water from a drain in the toilet, and what is the ruling on the prayers that were performed on it, knowing that it was not washed until after a long time, relying on the saying of Sheikh al-Islam Ibn Taymiyyah regarding the purity of the earth by dryness?

If you are certain that your shoe is impure and you walked with it while it was wet over pure places, there is a scholarly dispute regarding the purification of land by dryness, sun, and wind. This is the view of the Hanafis and one narration from Ahmad, chosen by Sheikh al-Islam. Alternatively, it is not purified except by water, which is the view of the majority and is more cautious. Both opinions are considered valid, and whoever adopts either of them is upon good. As for your prayer in this situation, if you were ignorant of or forgot the impurity, some scholars obliged its repetition, such as the Shafi'is and Hanbalis, and this is more cautious, while others did not. All of this applies if you are certain of the impurity that clung to your shoe; otherwise, the default is purity.