The meaning of $k \delta (t)$
You are correct. In your example, $k$ would refer to the "area underneath" the impulse. Mathematically speaking, the Dirac delta isn't a function in the typical sense of the term. Instead, it is more of a distribution, characterized by the fact that when integrated across any interval that contains $t=0$, the result is unity. That is:

$$
\int_{-\epsilon}^{\epsilon} \delta(t) dt = 1 \ \forall\ \epsilon &gt; 0
$$ 

The distribution is typically defined as follows:

$$
\delta(t) = \begin{cases} \infty,\ t = 0 \\ 0, \text{ otherwise}\end{cases}
$$

Multiplication by a constant $k$ obviously wouldn't change this definition at all. So instead, the scaling merely changes the area underneath the distribution:

$$
\int_{-\epsilon}^{\epsilon} k\delta(t) dt = k \ \forall\ \epsilon &gt; 0
$$

Extending this concept of multiplication by a constant $k$ to multiplication by a function $f(t)$ yields the following:

$$
\int_{-\epsilon}^{\epsilon} f(t) \delta(t) dt = f(0) \ \forall\ \epsilon &gt; 0
$$

or more generally:

$$
\int_{T-\epsilon}^{T+\epsilon} f(t) \delta(t-T) dt = f(T) \ \forall\ \epsilon &gt; 0
$$

which is known as the sifting property of the Dirac delta and is used extensively in linear systems theory.