Step 1: Note what is given.
The force on a body follows the rule $F = k v^{n}$, where $k$ is a fixed number and $n$ is an unknown power. We must find $n$ so that the power stays the same for every speed.
Step 2: Recall the link between power and force.
Power is the rate of doing work. For a force pushing along the motion, power is simply force times speed: $P = F v$. This is a basic result we can always use.
Step 3: Put the given force into the power formula.
Replace $F$ with $k v^{n}$, so $P = (k v^{n}) \times v$.
Step 4: Combine the powers of v.
When we multiply $v^{n}$ by $v$, we add the exponents. This gives $P = k v^{n+1}$.
Step 5: Use the key condition.
The problem says power does not depend on speed. The only way $k v^{n+1}$ can be free of $v$ is if the exponent is zero, because $v^{0} = 1$ for any speed.
Step 6: Solve for n.
Set the exponent to zero: $n + 1 = 0$, so $n = -1$. This makes $P = k$, a constant. \[ \boxed{-1} \]