To solve the given equation:

\[
\left(\frac{\sqrt{x}}{\sqrt{x} - 1} - \frac{1}{x - \sqrt{x}}\right) : \left(\sqrt{x} + \frac{1}{\sqrt{x}} + 2\right) = 2
\]

we need to simplify and solve step by step. Let's start by simplifying the left-hand side of the equation.

First, simplify the expression inside the parentheses:

\[
\frac{\sqrt{x}}{\sqrt{x} - 1} - \frac{1}{x - \sqrt{x}}
\]

To combine these fractions, we need a common denominator. The common denominator is \((\sqrt{x} - 1)(x - \sqrt{x})\).

Rewrite each fraction with the common denominator:

\[
\frac{\sqrt{x} \cdot (x - \sqrt{x})}{(\sqrt{x} - 1)(x - \sqrt{x})} - \frac{1 \cdot (\sqrt{x} - 1)}{(\sqrt{x} - 1)(x - \sqrt{x})}
\]

Simplify the numerators:

\[
\frac{x\sqrt{x} - x}{(\sqrt{x} - 1)(x - \sqrt{x})} - \frac{\sqrt{x} - 1}{(\sqrt{x} - 1)(x - \sqrt{x})}
\]

Combine the fractions:

\[
\frac{x\sqrt{x} - x - (\sqrt{x} - 1)}{(\sqrt{x} - 1)(x - \sqrt{x})}
\]

Simplify the numerator:

\[
x\sqrt{x} - x - \sqrt{x} + 1
\]

So the expression becomes:

\[
\frac{x\sqrt{x} - x - \sqrt{x} + 1}{(\sqrt{x} - 1)(x - \sqrt{x})}
\]

Now, consider the second part of the left-hand side:

\[
\sqrt{x} + \frac{1}{\sqrt{x}} + 2
\]

Combine the entire left-hand side:

\[
\frac{\frac{x\sqrt{x} - x - \sqrt{x} + 1}{(\sqrt{x} - 1)(x - \sqrt{x})}}{\sqrt{x} + \frac{1}{\sqrt{x}} + 2}
\]

This simplifies to:

\[
\frac{x\sqrt{x} - x - \sqrt{x} + 1}{(\sqrt{x} - 1)(x - \sqrt{x})(\sqrt{x} + \frac{1}{\sqrt{x}} + 2)} = 2
\]

To solve this equation, we need to find \(x\) such that the left-hand side equals 2. This is a complex algebraic expression, and solving it analytically might be challenging. Instead, we can use numerical methods or graphing techniques to find the value of \(x\).

However, let's check if there are any obvious solutions by substituting simple values for \(x\):

1. \(x = 1\):
\[
\left(\frac{\sqrt{1}}{\sqrt{1} - 1} - \frac{1}{1 - \sqrt{1}}\right) : \left(\sqrt{1} + \frac{1}{\sqrt{1}} + 2\right) = 2
\]
This leads to a division by zero, so \(x = 1\) is not a solution.

2. \(x = 4\):
\[
\left(\frac{\sqrt{4}}{\sqrt{4} - 1} - \frac{1}{4 - \sqrt{4}}\right) : \left(\sqrt{4} + \frac{1}{\sqrt{4}} + 2\right) = 2
\]
Simplifying:
\[
\left(\frac{2}{2 - 1} - \frac{1}{4 - 2}\right) : \left(2 + \frac{1}{2} + 2\right) = 2
\]
\[
\left(2 - \frac{1}{2}\right) : \left(2 + 0.5 + 2\right) = 2
\]
\[
\left(\frac{4}{2}\right) : 4.5 = 2
\]
\[
2 : 4.5 = 2
\]
This does not hold true.

Therefore, solving this equation analytically is complex, and we might need to use numerical methods or graphing techniques to find the exact solution for \(x\).