Praise be to Allah.
With regard to wiping over the khuffayn or socks, it is stipulated that the sock should cover the place that must be washed, and the place that must be washed goes from the ends of the toes to the ankles.
Ibn Qudaamah (may Allah have mercy on him) said: It is permissible to wipe over the khuffayn and the like if it covers the place that must be washed. If any part of the place that must be washed is not covered, then it is not permissible to wipe over it.
End quote from al-Mughni, 1/183
Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him) said: It is not allowed to wipe over the socks and so on unless the conditions stipulated in sharee‘ah are met, one of which is that the socks should cover the place that must be washed.
End quote from Majmoo‘ Fataawa Ibn Baaz, 10/114
Based on that:
The socks mentioned in the question do not come under the ruling on wiping over the khuffayn, because they do not cover the place that must be washed. But if the sick person needs to wear them and it is difficult for him to take them off every time he does wudoo’, then he may do wudoo’ before putting them on, and then put over them socks that cover properly. Then when he needs to do wudoo’, he can wipe over the socks that are covering them, until he takes them off or the period during which wiping over the socks is permissible ends. This is something acceptable and there is nothing wrong with it.
If he is not able to do that, and taking off the medical sock is something that will cause him harm or delay his recovery, then it comes under the same ruling as plaster casts; he may wipe over the entire thing, top and bottom – and this is one of the differences between wiping over plaster casts and wiping over the khuffayn: the entire cast should be wiped, whereas only the top of the khuffayn is wiped – and the uncovered part of the foot should be washed.
For more information please see the answer to question no. 114192
What your mother should have done was ask about what was required in that situation. The fact that she did not ask indicates that she was careless; we ask Allah to forgive her.
With regard to making up the prayers that she offered in that situation, the scholars (may Allah have mercy on them) differed concerning that. Some of the scholars say that the prayers must be made up and some say that it is not obligatory to do that.
If she wants to be on the safe side and make up the prayers that she offered with that wudoo’, that will be better.
If it is too difficult for her to do that or the days on which she offered prayers in that state were many, then she may be excused for not knowing, if she thought that what she was doing was correct and it did not occur to her to ask about it. In that case she does not have to repeat them according to the more correct of the two scholarly opinions.
And Allah knows best.