Is it permissible to pay Zakat on money in installments, and can the cost of using the internet service belonging to the Cancer Hospital - the proceeds of which go entirely to the hospital - be counted as part of Zakat?

It is not permissible to delay the payment of Zakat (definition) without a valid excuse; rather, it becomes immediately obligatory once it is possible to pay it. Zakat should not be given to every sick person; instead, the sick person must be among the eight categories mentioned in the verse from Surat At-Tawbah (definition): "Indeed, Charity is for the poor, the needy, those employed to collect [Zakat], those whose hearts are to be reconciled [to Islam], for freeing captives, for those in debt, for the cause of Allah, and for the wayfarer - an obligation [imposed] by Allah. And Allah is All-Knowing, All-Wise."