Determine the first term (a) and the common difference (d) of the arithmetic progression.

The $k$th term is given by $u_k = 89 - 5k$.
To find the first term, set $k=1$:

$$u_1 = 89 - 5(1) = 89 - 5 = 84$$

So, the first term $a = 84$.

The common difference $d$ is the coefficient of $k$ in the expression for $u_k$, which is $-5$.
Alternatively, find the second term and subtract the first term:

$$u_2 = 89 - 5(2) = 89 - 10 = 79$$ $$d = u_2 - u_1 = 79 - 84 = -5$$

The sum of the first $n$ terms, $S_n$, is given by the formula:

$$S_n = \frac{n}{2}(2a + (n-1)d)$$

We are given $S_n = -35$, $a = 84$, and $d = -5$. Substitute these values into the formula:

$$-35 = \frac{n}{2}(2(84) + (n-1)(-5))$$

$$-35 = \frac{n}{2}(168 - 5n + 5)$$ $$-35 = \frac{n}{2}(173 - 5n)$$

Multiply both sides by 2:

$$-70 = n(173 - 5n)$$ $$-70 = 173n - 5n^2$$

Rearrange the terms to form a quadratic equation:

$$5n^2 - 173n - 70 = 0$$

Here, $A=5$, $B=-173$, and $C=-70$.

$$n = \frac{-(-173) \pm \sqrt{(-173)^2 - 4(5)(-70)}}{2(5)}$$ $$n = \frac{173 \pm \sqrt{29929 + 1400}}{10}$$ $$n = \frac{173 \pm \sqrt{31329}}{10}$$

Calculate the square root:

$$\sqrt{31329} = 177$$

Substitute this back into the equation for $n$:

$$n = \frac{173 \pm 177}{10}$$

This gives two possible values for $n$:

$$n_1 = \frac{173 + 177}{10} = \frac{350}{10} = 35$$ $$n_2 = \frac{173 - 177}{10} = \frac{-4}{10} = -0.4$$

Since $n$ must be a positive integer representing the number of terms, we discard $n = -0.4$.

The value of $n$ is $\boxed{\text{35}}$.