jmc

Let Earth's radius be $r$. Since the equator measures 25100 miles, we have $2\pi r=25100\Rightarrow r=\frac{12550}{\pi }$.



Let Earth's center be $A$, let the center of the circle that passes through Lena be $B$, and let Lena be $L$. Because $\overline{BL}$ is parallel to the equator and Lena is at $45^{\circ }$ North Latitude, $△ABL$ is a 45-45-90 triangle. Thus, $BL=AB=\frac{r}{\sqrt{2}}$.

The number of miles in the circumference of the circle parallel to the equator and through Lena is $2\pi \cdot BL=2\pi \frac{r}{\sqrt{2}}=\frac{25100}{\sqrt{2}}\approx 17748$ miles. To the nearest hundred miles, this value is $\boxed{17700}$ miles.