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 $S$-number because $\sqrt{81}=8+1$.
$6724$ is an $S$-number: $\sqrt{6724}=6+72+4$.
$8281$ is an $S$-number: $\sqrt{8281}=8+2+81=82+8+1$.
$9801$ is an $S$-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 $T({10}^{12})$.