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 (LHS) of the equation.

First, consider 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 for \(\sqrt{x} - 1\) and \(x - \sqrt{x}\) is \((\sqrt{x} - 1)(x - \sqrt{x})\).

Rewrite each fraction with the common denominator:

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

Simplify the numerators:

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

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 entire LHS of the original equation:

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

This can be rewritten as:

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

Since the entire expression equals 2, we have:

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

To solve this equation, we need to equate the numerator to twice the denominator:

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

This equation is quite complex and solving it analytically might be challenging. However, we can check for specific values of \(x\) to see if they satisfy the equation. Let's try \(x = 1\):

For \(x = 1\):

\[
\sqrt{1} = 1
\]

Substitute \(x = 1\) into the original equation:

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

This results in a division by zero, which is undefined. Therefore, \(x = 1\) is not a solution.

Let's try \(x = 4\):

For \(x = 4\):

\[
\sqrt{4} = 2
\]

Substitute \(x = 4\) into the original equation:

\[
\left(\frac{2}{2 - 1} - \frac{1}{4 - 2}\right) : \left(2 + \frac{1}{2} + 2\right) = 2
\]

Simplify:

\[
\left(\frac{2}{1} - \frac{1}{2}\right) : \left(2 + 0.5 + 2\right) = 2
\]

\[
\left(2 - 0.5\right) : 4.5 = 2
\]

\[
\frac{1.5}{4.5} = 2
\]

\[
\frac{1}{3} = 2
\]

This is not true, so \(x = 4\) is not a solution either.

Given the complexity of the equation, it might be more practical to use numerical methods or graphing techniques to find the solution. However, based on the algebraic manipulations, it seems there might be no simple solution for \(x\) that satisfies the given equation.