How does my mother pay Zakat on an amount of 6566 Omani Riyals, of which 2000 is invested in a quarry company with a monthly profit, and 1950 is saved for Hajj? Is Zakat calculated on the entire amount? What is the amount of Zakat and to whom is it given?

Zakat is obligatory on the entirety of your mother's wealth (the money she possesses, the invested money along with its profit if any, and the money saved for Hajj), as it all belongs to one person. If this wealth reaches the nisab and a full lunar year (hawl) has passed over it, then one-quarter of one-tenth (2.5%) of it is to be given as zakat. Zakat is to be distributed to the eight categories mentioned in the Holy Quran in Surah At-Tawbah: 60.