Optimal Production Input in Relation to Cost Minimization Problem
I`d propose you to follow these steps:


Set up the minimization cost problem (i.e. for a given output quantity $y$ minimize costs):
\begin{align}
\min_{H,L,K}&amp;  \quad sH + wL + rK \tag{1} \label{1}\\
\text{such that} &amp;\quad  \min\{H,L\} + \min\{H, K\}\geq y \tag{2} \label{2}
\end{align}
In principle you have 3 cases, depending on price of factors $(s,w,r)$.

1) When $r&gt;s+w$, then buying a unit of $L$ and one of $H$ is going to cost you strictly less than buying a unit ok $K$. Consequently the monopolist opts for producing only using $H$ and $L$ ($K=0$). Thus $H = L = y$ 


2) The second case happens for $s&gt;r+w$ which is, anyway, ruled out by the assupmption $w&gt;s$ (if this was not the case, then producing only using $H$ and $K$ would have been optimal, i.e $H = K = y$ and $L = 0$). 

3) Finally, when none of the above occurs, you have $L = H$ and $K = H$ because of the fixed proportions production function. Then the constraint \eqref{2} becomes $H + H \geq y$ and the inequality sign become equality. The equality sign comes from the fact that producing more than $y$ (the LHS of \ref{2} being strictly larger than the RHS) is just overshooting and inflates production costs for a given $y$. $L = H$ and $K = H$ comes from the Leontief production set-up of the problem: inside the $\min\{...,...\}$ operator is never convenient to exploit more one of the two factors (because by decreaing the most used factor one still gets the same amount of output but a lower cost). Therefore, each of the two $\min\{...,...\}$ can just be seen as one of its inner arguments, namely, for convenience, $H$. So you finally get $H+H = y$ and, from there: $H = y/2$ and $L = K = y/2$


Substitute the optimal values ${\big(H^*\small{(s,w,r)},L^*\small{(s,w,r)},K^*\small{(s,w,r)}\big)}$ in \eqref{1} so to get an expression only in $(s,w,r)$. This expression is the cost function $C(s,w,r,y)$
Now solve the monopolist's pricing problem for given factor prices:
\begin{equation}
\max_p \quad p\cdot D(p) - C\big(s,w,r,D(p)\big)
\end{equation}
and find the profit maximizing price, $p^*$ for output level $y^* = D(p^*)$


I hope this helps, comment below if you need any more hint.