It is possible that you did not understand the mentioned references. Let say that $T_s$ is the time that is needed for completion of each stage, and $k$ is the number of stages. $T_s$ may or may not include the cost of pipelining. The latency of the first instruction is equal to the sum of times needed for completion of each stage, i.e., $kT_s$. Once the pipeline is full, the latency of each subsequent instruction is $T_s$. Therefore, the total time needed for execution of $n$ instructions is: $kT_s$ (first instruction) $+ (n - 1) T_s$ (each subsequent instruction).
Therefore, when pipeline is full, latency of each instruction is $T_s$. Still, the time needed for completion of each instruction is $kT_s$. Please note the difference between the latency and the time needed for completion of instruction. Pipelining only reduces the latency.
