The CDF of $K$ is $$ P(K \le n) = P(S_n > T) = \int_{T}^\infty dt\; f_n(t)$$ The CDF of $S_K$ (for $s > 0$) is $$\eqalign{P(S_K \le s) &= \sum_{n=1}^\infty P(K = n, S_n \le s) = \sum_{n=1}^\infty P(S_{n-1} \le T, T < S_n \le s)\cr &= \sum_{n=1}^\infty \int_0^T dt\; f_{n-1}(t) \int_T^s dr\; f(r-t)}$$

The CDF of $K$ is $$ P(K \le n) = P(S_n > T) = \int_{T}^\infty dt\; f_n(t)$$ The CDF of $S_K$ (for $s > T$) is $$\eqalign{P(S_K \le s) &= \sum_{n=1}^\infty P(K = n, S_n \le s) = \sum_{n=1}^\infty P(S_{n-1} \le T, T < S_n \le s)\cr &= \sum_{n=1}^\infty \int_0^T dt\; f_{n-1}(t) \int_T^s dr\; f(r-t)}$$

EDIT: the $n=1$ term needs to be modified since $S_0 = 0$ doesn't have a density. So (assuming of course $T > 0$) it's $$P(S_K \le s) = \int_T^s dr\; f(r) + \sum_{n=2}^\infty \int_0^T dt\; f_{n-1}(t) \int_T^s dr\; f(r-t)$$