Conditional distribution function for two standard normal variables

Let's focus on your original exponent:

$$-\frac{u^2+v^2-2\rho uv}{2(1-\rho^2)}$$

observe that we can manipulate it in the following manner

$$ \bbox[5px,border:2px solid black] { -\frac{u^2+v^2-2\rho uv\pm\rho^2v^2}{2(1-\rho^2)}=-\frac{v^2}{2}-\frac{(u-\rho v)^2}{2(1-\rho^2)} \qquad (1) } $$

substituing this new exponent in you original joint density you will prove what you are asked to do.

In fact you will find

$$f_{UV}(u,v)=\frac{1}{\sqrt{2\pi}}e^{-v^2/2}\times\frac{1}{\sqrt{2\pi}\sqrt{1-\rho^2}}e^{-(u-\rho v)^2/2(1-\rho^2)}=f_V(v)\cdot f_{U|V}(u|v)$$

where

$$V\sim N(0;1)$$

$$(U|V)\sim N(\rho v;1-\rho^2)$$

mean and variance of the conditional density are immediately shown in the second addend of the manipulated exponent in (1)