The answer is $k=100$. To begin with we will prove that we always can split a $100 \times 100$ table into several rectangles with $k \le 100$. Let's split this table into $100$ columns and every column we can split into rectangles increasing $k$ by no more than $1$. It remains for us to show an example where $k=100$. Mark cells with coordinates $( i , i+1 ), ( i+1 , i )$ for all $i$ from $1$ to $99$ and $( j , j )$ for all $j$ from $2$ to $99$. And if we apply a chess coloring then the rest isn't hard.