The study of quantum thermodynamics, i.e. equilibrium and non-equilibrium thermodynamics of quantum systems, has been applied to various many-body problems, including quantum phase transitions. An important question is whether out-of-equilibrium quantities from this emerging field, such as fluctuations of work, exhibit scaling after a sudden quench. In particular, it is very interesting to explore this problem in impurity models where the lack of an obvious symmetry breaking at criticality makes it very challenging to characterize. Here, by considering a spin emulation of the two impurity Kondo model and performing density matrix renormalization group computations, we establish that the irreversible work produced in a quench exhibits finite-size scaling at quantum criticality. Our approach predicts the equilibrium critical exponents for the crossover length and the order parameter of the model, and, moreover, implies a new exponent for the rescaled irreversible work.