Is the wife's mother required to pay the full cost of Hajj, and would her Hajj be valid if the husband paid for her Hajj expenses from his own money, given that she wishes to pay for the sacrifice (udhiyah) from her own money?

It is not a condition for the validity of the wife's mother's Hajj that the expenditure be from her own money. So, if she performs Hajj at your expense, or if part of it is from you, her Hajj is valid. And there is no harm if you pay her the value of the sacrificial animal (Hadi) or the value of the offering (Udhiyah).