The design of energy and spectrally efficient Wireless Sensor Networks (WSN) is crucial to support the upcoming expansion of IoT/M2M mobile data traffic. In this work, we consider an energy harvesting WSN where sensor data are periodically reported to a Fusion Center (FC) by a sparse set of active sensors. Unlike most existing works, the transmit power levels of each sensor are assumed to be unknown at the FC in this distributed setting. We address the inverse problem of joint signal / power restoration at the FC- a challenging under-determined separation problem. To regularize the ill-posed problem, we assume both a graph-signal smoothness prior (signal is smooth with respect to a graph modeling spatial correlation among sensors) and a sparsity power prior for the two unknown variables. We design an efficient algorithm by alternately fixing one variable and solving for the other until convergence. Specifically, when the signal is fixed, we solve for the power vector using Simplex pivoting in linear programming (LP) to iteratively identify sparse feasible solutions, locally minimizing an objective. Simulation results show that our proposal can achieve very low reconstruction errors and outperform conventional schemes under reasonable assumptions