How is zakat paid on a laundry business that has been in operation for a full year: Is it based on the value of the equipment or on the profits, given that the profits have only been used to cover expenses?

Zakat (definition) is not obligatory on laundry equipment. However, it is obligatory on the profits if a year has passed over them and they have reached the Nisab (definition). They are then subject to Zakat at a rate of 2.5%. As for what a year has not passed over or what has not reached the Nisab, there is no Zakat on it.