If $2^\gamma \leq \kappa$ for all $\gamma < \kappa$ (e.g., if GCH holds and $\kappa$ is the least $\aleph_1$-strongly compact), then $|j(\kappa)| \geq 2^\lambda$. To see this, let $\sigma = [\text{id}]$. By elementarily, $M\vDash j(\kappa) \geq |P^M(\sigma)|$. But there is an injection $i : P(\lambda)\to P^M(\sigma)$ given by $i(A) = j(A)\cap \sigma$, so $|P^M(\sigma)| \geq 2^\lambda$.

If $2^\gamma \leq \kappa$ for all $\gamma < \kappa$ (e.g., if GCH holds and $\kappa$ is the least $\aleph_1$-strongly compact), then $|j(\kappa)| \geq 2^\lambda$. To see this, let $\sigma = [\text{id}]$. By elementarity, $M\vDash j(\kappa) \geq |P^M(\sigma)|$. But there is an injection $i : P(\lambda)\to P^M(\sigma)$ given by $i(A) = j(A)\cap \sigma$, so $|P^M(\sigma)| \geq 2^\lambda$.