To solve this problem, we need to use the properties of a rectangle inscribed in a circle.

Let's start by drawing a diagram of the situation described in the problem:

```
A _______ B
| |
| |
D|_______|C
```

Given that the circle has a radius of 13 cm, we can draw the circle with center O and radius 13 cm. The four vertices of the rectangle, A, B, C, and D, lie on the circumference of the circle.

Now, let's consider the properties of a rectangle inscribed in a circle:

1. The diagonals of the rectangle are equal in length and are also diameters of the circle.
2. The diagonals of the rectangle bisect each other.

Using these properties, we can find the length and breadth of the rectangle.

Let's consider the diagonals of the rectangle. Since the diagonals of the rectangle are diameters of the circle, they have a length of 2 * radius = 2 * 13 = 26 cm.

Since the diagonals of the rectangle bisect each other, we can divide the diagonals into two equal parts. Let's consider the diagonal AC. We can divide AC into two equal parts, which we can call AE and EC.

Now, we have a right-angled triangle AEO, where EO is the radius of the circle (13 cm) and AO is half of the diagonal AC (AE + EO).

Using the Pythagorean theorem, we can find the length of AE. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, we have:
```
AE^2 + EO^2 = AO^2
AE^2 + 13^2 = (26/2)^2
AE^2 + 169 = 169
AE^2 = 0
AE = 0
```
But, we know that the length of AE cannot be zero, so this case is not possible.

Now, let's consider the other diagonal BD. Again, we can divide BD into two equal parts, which we can call BF and FD.

Using the same reasoning as before, we can find the length of BF. Again, using the Pythagorean theorem, we have:
```
BF^2 + EO^2 = BO^2
BF^2 + 13^2 = (26/2)^2
BF^2 + 169 = 169
BF^2 = 0
BF = 0
```
But, we know that the length of BF cannot be zero, so this case is also not possible.

Therefore, we conclude that there are no possible pairs of length and breadth for the rectangle.