With two variables, you are defining a line segment in $\mathbb{R}^2$, as you pointed out. However, due to the simplex constraint, one of these two variables is *redundant* in terms of specifying the density, since there is a one-to-one relationship between $x_1$ and $x_2$. Therefore, the density is specified over $K-1$ free variables (i.e., in $\mathbb{R}$)

This is actually pointed out in the first line of [this section][1] of the Wikipedia article, albeit *very* subtly.

Therefore, your density function becomes:

$$Dir_{1,1}(x_1,1-x_1)=\frac{\Gamma(2)}{\Gamma(1)^2}(x_1)^0(1-x_1)^0=1$$

Therefore, 

$$\int_0^1 Dir_{1,1}(x_1,1-x_1) dx_1 = 1$$


  [1]: https://en.wikipedia.org/wiki/Dirichlet_distribution#Probability_density_function