1. Set up the geometry

Let mirror $A$ be at $x=0$ and mirror $B$ be at $x=10\,\text{cm}$.

The object $O$ is $2\,\text{cm}$ from mirror $A$, so its coordinate is $x=2\,\text{cm}.$

We need the images behind mirror $A$, i.e. images with coordinate $x<0$.

2. First image behind mirror $A$

The direct image of the object in mirror $A$ is formed at equal distance behind it: $x=-2\,\text{cm}.$ So the nearest image behind mirror $A$ is at a distance $2\,\text{cm}.$

3. Find the next image behind mirror $A$

To get another image behind $A$, first reflect the object in mirror $B$.

Object is at $x=2$, mirror $B$ is at $x=10$, so its image in $B$ is at $x=2\cdot 10 - 2 = 18\,\text{cm}.$

Now this acts as an object for mirror $A$. Reflection in mirror $A$ changes $x \to -x$, so image forms at $x=-18\,\text{cm}.$

Thus the second nearest image behind mirror $A$ is at a distance $18\,\text{cm}$ from mirror $A$.

4. Answer

The required integer is $\boxed{18}.$