Given,

$\overrightarrow{a}=\lambda \hat{i}+\mu \hat{j}+4\hat{k},\overrightarrow{b}=-2\hat{i}+4\hat{j}-2\hat{k}$ and $\overrightarrow{c}=2\hat{i}+3\hat{j}+\hat{k}$ are coplanar 

So, $\left|\begin{array}{ccc}\lambda & \mu & 4\\ -2& 4& -2\\ 2& 3& 1\end{array}\right|=0$

$\Rightarrow \lambda \left(10\right)-\mu \left(2\right)+4(-14)=0$

$\Rightarrow 10\lambda -2\mu =56$

$\Rightarrow 5\lambda -\mu =28 .....\left(1\right)$

And given projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ is $\sqrt{54}$

$\Rightarrow \frac{\overrightarrow{\mathrm{a}}·\overrightarrow{\mathrm{b}}}{\left|\overrightarrow{\mathrm{b}}\right|}=\sqrt{54}$

$\Rightarrow \frac{-2\lambda +4\mu -8}{\sqrt{24}}=\sqrt{54}$

$\Rightarrow -2\lambda +4\mu -8=\sqrt{54\times 24} ....\left(2\right)$

By solving equation $\left(1\right) \& \left(2\right)$ we get,

$\Rightarrow \lambda +\mu =24$