Limits and Functional equations!

Relevant wiki: Differentiation Under the Integral Sign

$\color{#D61F06} \text{To Find:}$ $\color{#20A900} \displaystyle{\lim _{ x\rightarrow \frac { \pi }{ 4 }}{ \left( \frac { \displaystyle \int _{ 2 }^{ \sec ^{ 2 }{ x } }{ f\left( t \right) dt } }{ { x }^{ 2 }-\dfrac { { \pi }^{ 2 } }{ 16 } } \right) } =?}$

Now We can Apply L'Hospitals Rule and Leibnitz Integral Rule

We Have

$\implies \displaystyle{\lim _{ x\rightarrow \frac { \pi }{ 4 } }{ \left( \frac { \displaystyle \int _{ 2 }^{ \sec ^{ 2 }{ x } }{ f\left( t \right) dt } }{ { x }^{ 2 }-\dfrac { { \pi }^{ 2 } }{ 16 } } \right) }}$

$\implies \displaystyle{\lim_{x\rightarrow \frac{\pi}{4}}{\left(\frac{2 \times \sec^2x \times \tan x \times f(sec^2x)}{2x}\right)}}$ $\small{\color{#3D99F6} \text{Differentiated and then used Leibnitz Integral Rule}}$

Putting the value of the limits,

$\implies \large{ \frac{ \tan \dfrac{\pi}{4} \times \sec^2 \dfrac{\pi}{4} \times \sec^2 x\times f(\sec^2 \dfrac{\pi}{4})}{2 \times \dfrac{\pi}{4}}}$

$\implies \large{ \frac{ 1 \times 2 \times 2 \times f(2)}{\dfrac{\pi}{2}}}$

$\implies \large{\frac{8f(2)}{\pi}}$

so $k^a = 8^2 \implies \boxed{64}$