To simplify the given expression, let's break it down step by step:

\[ \frac{1}{\sqrt{5} - 2} + \frac{1}{\sqrt{5} + 2} + 2024 - \frac{\sqrt{2024}}{1 - \sqrt{2024}} \]

### Step 1: Simplify the first two fractions

For the first fraction \(\frac{1}{\sqrt{5} - 2}\), we rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{5} + 2\):

\[ \frac{1}{\sqrt{5} - 2} \cdot \frac{\sqrt{5} + 2}{\sqrt{5} + 2} = \frac{\sqrt{5} + 2}{(\sqrt{5} - 2)(\sqrt{5} + 2)} = \frac{\sqrt{5} + 2}{5 - 4} = \frac{\sqrt{5} + 2}{1} = \sqrt{5} + 2 \]

For the second fraction \(\frac{1}{\sqrt{5} + 2}\), we rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{5} - 2\):

\[ \frac{1}{\sqrt{5} + 2} \cdot \frac{\sqrt{5} - 2}{\sqrt{5} - 2} = \frac{\sqrt{5} - 2}{(\sqrt{5} + 2)(\sqrt{5} - 2)} = \frac{\sqrt{5} - 2}{5 - 4} = \frac{\sqrt{5} - 2}{1} = \sqrt{5} - 2 \]

Now, add the simplified fractions:

\[ \sqrt{5} + 2 + \sqrt{5} - 2 = 2\sqrt{5} \]

### Step 2: Simplify the third term

The third term is simply \(2024\).

### Step 3: Simplify the fourth term

For the fourth term \(\frac{\sqrt{2024}}{1 - \sqrt{2024}}\), we rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator, which is \(1 + \sqrt{2024}\):

\[ \frac{\sqrt{2024}}{1 - \sqrt{2024}} \cdot \frac{1 + \sqrt{2024}}{1 + \sqrt{2024}} = \frac{\sqrt{2024}(1 + \sqrt{2024})}{(1 - \sqrt{2024})(1 + \sqrt{2024})} = \frac{\sqrt{2024} + 2024}{1 - 2024} = \frac{\sqrt{2024} + 2024}{-2023} = -\frac{\sqrt{2024} + 2024}{2023} \]

### Step 4: Combine all terms

Now, we combine all the simplified terms:

\[ 2\sqrt{5} + 2024 - \frac{\sqrt{2024} + 2024}{2023} \]

This is the simplified form of the given expression.