How to calculate the Fourier Transform of a constant?

Your computation is incorrect, because the value of $e^{-j2\pi f t}$ oscillates around the unit circle in the complex plane as $t \to \pm \infty$ (it doesn't approach either $0$ or $\infty$), unless $f = 0$, in which case it is constantly equal to $1$.

Thus, if we average over all values of $t$, we get $0$ if $f \neq 0$ (all the oscillations in the different directions cancel out), while we get $\infty$ if $f = 0$. So we have a function which is zero at all $f \neq 0$ and infinite at $f = 0$, i.e. $\delta(f)$. If we take the initial constant to be $1/2$ instead of $1$, we get $\frac{1}{2} \delta(f)$, as you surmise.

(This is a slightly informal discussion. The correct mathematical formalism for handling $\delta(f)$ is the theory of distributions. It's quite likely that someone else will post an answer, or maybe a link, discussing this more formally correct approach.)