To solve the 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.

First, let's rewrite the equation for clarity:

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

This can be interpreted as:

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

To solve this, we need to simplify the numerator and the denominator separately.

### Simplifying the Numerator

The numerator is:

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

First, let's find a common denominator for the two fractions:

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

Simplify the numerator:

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

So the numerator becomes:

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

### Simplifying the Denominator

The denominator is:

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

Combine the terms:

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

### Putting It All Together

Now, we have:

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

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 equate the numerator and the denominator to the right-hand side:

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

This equation is quite complex and might require numerical methods or further algebraic manipulation to solve. However, let's check if there are any obvious solutions by substituting some values for \(x\).

For example, if \(x = 1\):

\[
\sqrt{1} = 1
\]

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, so \(x = 1\) is not a solution.

Let's try another value, \(x = 4\):

\[
\sqrt{4} = 2
\]

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:

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

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

\[
1.5 : 4.5 = 2
\]

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

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

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

Given the complexity, it might be best to use numerical methods or graphing techniques to find the exact solution.