Let's break down the problem and solution step by step.

### Part (a): Side Length of the Middle Square

1. Understanding the Sequence of Squares:
- The squares are placed such that their midpoints lie on the diagonal of the unit square.
- The side length of each subsequent square is either half or double the side length of the previous square.
- The distance from the midpoint of each square to the vertex of the unit square follows the sequence $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots$ of the diagonal.

2. Distance Calculation:
- The distance from the midpoint of the first square to the vertex of the unit square is $\frac{1}{2}$ of the diagonal.
- The diagonal of the unit square is $\sqrt{2}$.
- Therefore, the distance from the midpoint of the first square to the vertex is $\frac{\sqrt{2}}{2}$.

3. Side Length Calculation:
- Let the side length of the first square be $k$.
- The distance from the midpoint of the first square to the vertex is $\frac{k\sqrt{2}}{2}$.
- Equating this to the distance calculated above:
$$\frac{k\sqrt{2}}{2} = \frac{\sqrt{2}}{2}$$
- Solving for $k$:
$$k = 1$$

### Part (b): Total Gold-Plated Area

1. Area Calculation:
- The area of the first square is $k^2$.
- The area of the second square (with side length $\frac{k}{2}$) is $\left(\frac{k}{2}\right)^2 = \frac{k^2}{4}$.
- The area of the third square (with side length $\frac{k}{4}$) is $\left(\frac{k}{4}\right)^2 = \frac{k^2}{16}$.
- This forms a geometric series with the first term $k^2$ and common ratio $\frac{1}{4}$.

2. Sum of the Series:
- The sum of the infinite geometric series is given by:
$$S = k^2 \left(1 + \frac{1}{4} + \frac{1}{16} + \cdots \right)$$
- The sum of the series is:
$$S = k^2 \left(\frac{1}{1 - \frac{1}{4}}\right) = k^2 \left(\frac{4}{3}\right)$$

3. Total Gold-Plated Area:
- Since the series covers half the diagonal, the total gold-plated area is twice this sum minus the center square:
$$\text{Total Area} = 2 \left(\frac{4}{3} k^2\right) - k^2 = \frac{8}{3} k^2 - k^2 = \frac{5}{3} k^2$$
- Substituting $k = 1$:
$$\text{Total Area} = \frac{5}{3} \times 1^2 = \frac{5}{3}$$

The final answer is: