How can batteries reach equilibrium if the reactions are irreversible?
Your confusion probably stems from the definition of a reversible path between two states: an infinite sequence of steps along a continuum of equilibrium states, such that at each step the equivalence condition of the second law of thermodynamics, $\mathrm dS_\text{total}=0$, is satisfied. 

However, there is no requirement that a path between two states must be reversible. Many textbook problems are based on comparing the entropy changes associated with reversible and irreversible paths. For the reversible paths, $\Delta S_\text{total}=0$, for the (spontaneous) irreversible ones, $\Delta S_\text{total}\gt0$.

Your textbook explains that the discharging process of a Daniell cell is considered irreversible. The usage of the term irreversible in the book is rather similar to the common usage, which means "impossible to bring back to its original state, for practical reasons.". The answer that Karl has given you is correct. Basically it is not possible to bring the cell to its original state. The chemical and physical integrity of the recharged electrodes and electrolyte are altered compared to the original structure. There are entropic costs associated with cycling the cell.