What is the sum of all $k\leq25$ such that one can completely cover a $k\times k$ square with $T$ tetrominos (shown in the diagram below) without any overlap?

[asy]
size(2cm);
draw((0,0)--(3,0));
draw((0,1)--(3,1));
draw((1,2)--(2,2));
draw((0,0)--(0,1));
draw((1,0)--(1,2));
draw((2,0)--(2,2));
draw((3,0)--(3,1));
[/asy]

$\text{(A) }20\qquad\text{(B) }24\qquad\text{(C) }84\qquad\text{(D) }108\qquad\text{(E) }154$