Timing of Nature Drawing Types
I suppose you have in mind a game that looks like this: 




  she knows her type when she is called upon to move, so why do we calculate an expected payoff for player 2 when converting the game into an normal-form?


The fact that "she knows her type" and therefore has no need to calculate an expected payoff presupposes that the play has reach a particular information set for player 2. Formally, let $\theta\in\{R,B\}$ be P2's type. Her utility  conditional on her type being $\theta$ is
\begin{equation}
u_2(\sigma_1,\beta_2(\color{red}{\theta})\vert\color{red}{\theta}). 
\end{equation}
where $\beta_2(\theta)$ is P2's behavior strategy at the information set associated with $\theta$ (i.e. the left decision node for P2 when $\theta=R$ and the right decision node for P2 if $\theta=R$). Here, no expected value is calculated since the uncertainty is resolved for P2.
In contrast, in the normal form representation, P2's utility is an unconditional one: 
\begin{equation}
u_2(\sigma_1,\sigma_2)=\Pr(R)u_2(\sigma_1,\beta_2(R)\vert R)+\Pr(B)u_2(\sigma_1,\beta_2(B)\vert B)
\end{equation}
where expectation is to be taken.

The reason is that by converting an extensive form game into the normal form, the temporal aspect of the game is lost and everything is seen from the ex ante perspective.