Number Splitting

We define an $S$-number to be a natural number, $n$, that is a perfect square and its square root can be obtained by splitting the decimal representation of $n$ into $2$ or more numbers then adding the numbers.

For example, $81$ is an -number because $\sqrt{81}=8+1$.
$6724$ is an -number: $\sqrt{6724} = 6 + 72 + 4$.
$8281$ is an -number: $\sqrt{8281} = 82 + 8 + 1$.
$9801$ is an -number: $\sqrt{9801} = 98 + 0 + 1$.

Further we define $T(N)$ to be the sum of all $S$ numbers $n\le N$. You are given $T(10^4) = 41333$.

Find $T(10^{12})$