Does she have to give zakah on the rent of the house?

Yes, zakah must be given on the rent that the wife receives, if one full year has passed. The year is to be counted from the time when the rental contract is drawn up, and at the end of that year she must give zakah on it or on what remains of it, if it reaches the minimum threshold (nisab), at a rate of 2.5%. If she has spent it all and has not saved anything of it, then no zakah is due on it.

Shaykh Ibn ‘Uthaymin was asked, as it says in Majmu‘ al-Fatawa (18/question no. 110):

What is the ruling on zakah on money that a person receives for apartments that he rents out, as he does not receive the rent for one apartment all in one go; rather he receives it in two or three payments?

He replied:

All the rent that a person receives little by little, if he spends it when he receives it, no zakah is due on it, so long as one year has not passed since the rental contract was drawn up.

One example of that is a man who rents out an apartment for ten thousand, and at the end of the year he receives ten thousand. In this case he must give zakah on it, because one year has passed since the contract was drawn up.

Another man received the rent in full in advance – meaning that the landlord received it at the time of drawing up the contract, then he spent it. In this case no zakah is due on it, because one full year did not pass since it was received, and one of the conditions of zakah being obligatory is that one full year should have passed.

As for the apartment itself, it is not subject to zakah, because anything that is prepared for rental is not subject to zakah, such as real estate, cars,  equipment and so on, except jewellery made of gold or silver, which is subject to zakah every year in all cases if it reaches the minimum threshold (nisab). End quote.

And Allah knows best.