The Eschenauer--Gligor (EG) random key predistribution scheme has been widely recognized as a typical approach to secure communications in wireless sensor networks (WSNs). However, there is a lack of precise probability analysis on the reliable connectivity of WSNs under the EG scheme. To address this, we rigorously derive the asymptotically exact probability of $k$-connectivity in WSNs employing the EG scheme with unreliable links represented by independent on/off channels, where $k$-connectivity ensures that the network remains connected despite the failure of any $(k-1)$ sensors or links. Our analytical results are confirmed via numerical experiments, and they provide precise guidelines for the design of secure WSNs that exhibit a desired level of reliability against node and link failures.