No such thing as a single frequency of the noise. That's exactly why it's called white; it has power in all frequency ranges, but not at a single frequency.
Finally, is there a frequency-domain representation of Gaussian white noise?
Yes, a constant power spectral density for all frequencies. That's like white light (which contains also a continuum of all frequencies), which is where it got it name from.
Dealing with continuous spectra, like that of white noise, but also those of actual signals is absolutely bread and butter of control engineering – we very rarely have an input to a system that's always constant or just a sinusoid, which means that its spectrum is never the actual one tone you mention in your question.
I'm not quite sure how to proceed here – I can't really paste three thirds of "Otto Föllinger – Regelungstechnik" (my German standard control engineering textbook) as an answer, and I'm pretty sure that would also not be super helpful to you – because probably, you'll touch upon how you can work with signals and systems in Laplace or Fourier domain, where this really becomes easier.
I'll thus venture a quick "outlook":
Your control system is linear (nod here, please, if you agree, and if you're not certain what linearity is, look it up), thus, if you input a sum of two different signals, the output is equal to the sum of outputs you would have gotten if you used them individually.
Thus, you can start approximating your white noise. Noise has a measurable power. So, pick one arbitrary frequency and put a tone of the same power here. A single line in frequency domain is not a good approximation of a noise that has no lines but is a constant "floor", but at least we've got the same power.
To make the approximation better, you now split the power (i.e. half it, which means multiplying the amplitude with $\frac1{\sqrt2}$) and make two tones that divide the frequency span you're looking at evenly. You calculate the output of your system for both inputs, add it, and get a possible output.
Small problem: You need to assume some phase of your tones. Well, noise can (and will) have independent random phase at different frequencies, so you calculate the output power for a large set of uniformly chosen random phases. You calculate the average power and claim it's an estimate for the output power frequency distribution for white noise.
Instead of halving the power and giving it to two tones, you could of course also had divided the power by 1,000,000 and given it to 1,000,000 tones. Same principle applies. Your estimate of the output power distribution will probably be much better than with 2 strong tones, since 1,000,000 weak tones with random phases are much more alike white noise than 2. However, your math senses are tingling intensely, and you recognize that you're basically doing a large $\sum\limits_f$ over 1,000,000 frequencies and really, you need to do it over uncountably many frequencies, and that sounds like a job for an $\int\cdots\,\mathrm df$.
Thus, you end up with an integral transform, such as the Laplace or Fourier transform. With these, you can describe signals that are deterministic, but have a continuous spectrum, which your Bode plot method seemingly cannot! Think about it: What is the "frequency" of "Input is 0 for eternity, until it turns 6 for a duration of 2, after which it's 0 again for eternity"? There is no single frequency in aperiodic things!
(and by the way, that's why I wrote it has no power at a single frequency: you divide a finite amount of power by an uncountably infinite amount of frequencies. Every frequency gets exactly 0 power. If you integrate over a range of frequencies, even an extremely small one, you get some power, so the result of your division is some kind of power density!)
The fact that the input is stochastic means you're in trouble if you assume fixed phase relationships, and that's why we usually have to resort to power spectral densities. However, from a simple "how does the output power look like if I feed white noise into a low-pass RC filter with a bode plot that looks like ...", you'll see that this is a very useful info to have!
Don't be afraid: The math you've learned for single-frequency inputs is absolutely necessary to build the continuous spectrum understanding, and it will be used all over the place in your continued studies of control engineering (or in fact, all disciplines of electrical engineering, if that's what you're studying). More often than not, the problem really boils down to "well, then just multiply the Bode plot point-wise with the input spectrum and sum / integrate things up"; and that's not really hard to do.
