What is the meaning of the DTFT of the unit impulse sequence?
Your calculation is correct.

Your signal has its energy equally distributed in all frequencies.

But note that in discrete-time, "all frequencies" means from $\omega = -\pi$ to $\omega = +\pi$. Frequencies beyond that range exist but result in the same signal as one with a frequency within the range.

For example, $\cos(0\times n) = \cos(2\pi \times n)$. Similarly $\cos(0.1 \pi \times n) = \cos(2.1 \pi \times n)$, and so on.

So the energy of the signal is calculated integrating in a range of $2\pi$.

The formal result is known as Parseval's theorem, which states that energy measured in time domain is equal to energy measured in frequency domain:

$\sum_n |x[n]|^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi} |X(e^{j\omega})|^2 d\omega$

Note: this is a particular version of Parseval. The more general version states that the projection (think cross product) of one vector onto another is the same calculated in time domain or frequency domain, which is reasonable since Fourier can be seen as a change of base, and a change of base should not alter projections.