Sensing coverage is an essential functionality of wireless sensor networks (WSNs). However, it is also well known that coverage alone in WSNs is not sufficient, and hence network connectivity should also be considered for the correct operation of WSNs. In this paper, we address the problem of k-coverage in WSNs such that in each scheduling round, every location in a monitored field (or simply field) is covered by at least k active sensors while all active sensors are being connected. Precisely, we study sensors duty-cycling strategies for generating k-coverage configurations in WSNs. First, we model the k-coverage problem in WSNs. Second, we derive a sufficient condition of the sensor spatial density for complete k-coverage of a field. We also provide a relationship between the communication and sensing ranges of sensors to maintain both k-coverage of a field and connectivity among all active sensors. Third, we propose four configuration protocols to solve the problem of k-coverage in WSNs. We prove that our protocols select a minimum number of sensors to achieve full k-coverage of a field while guaranteeing connectivity between them. Then, we relax some widely used assumptions for coverage configuration in WSNs, to promote the use of our proposed protocols in real-world sensing applications. Our simulation results show that our protocols outperform an existing distributed k-coverage configuration protocol.