Answer: 704

Small Hint:

After $8$ has been typed, any remaining letters among $1,\ldots,7$ must later be typed in decreasing order

Big Hint:

Separate the cases according to whether $9$ was typed before lunch or remains to be inserted into the later order

Solution:

Any subset of $1,\ldots,7$ can remain in the stack after $8$ has been typed, and those remaining letters must later be typed in decreasing order. If $9$ was already typed, choosing that subset gives $2^7=128$ possible orders.

If $9$ remains, choose $k$ of the seven smaller letters and insert $9$ into any of the $k+1$ positions in their decreasing order. This gives $$\begin{aligned}\sum_{k=0}^7\binom7k(k+1)&=7\cdot2^6+2^7\\&=576.\end{aligned}$$ Every such order can be realized by suitable morning choices, so the total is $128+576=704.$