I'm not afraid to respond. Harrumph.
You would probably be better off with pink noise than white noise. When measured in "real world" terms that are more realistic for the way we create and respond to sound, on a log scale, white noise gets louder as the frequency goes up. Wimpy midrange, and no lows.
Pink noise, on the other hand, is nicely balance compared to most sounds. It's how we tune sound systems in a room sometimes; more musically representative.
The main point is that for audio masking to work, the frequency spectrum and the amplitudes of the source and the masking have to be at least in the same ballpark.
Now... I don't even want to speculate about the particular sounds you'd like to mask, but to do it well enough to actually work, you should probably start with spectral analysis of, eh, uh... the actual sounds. Uhm.
Then you need to generate something like that spectrum at that amplitude.
The simple approach is just to shotgun it - a pink noise generator, an appropriate amp and speaker, a little experimenting, and hope for the best.
It occurs to me that the levels will be lower and more representative at the, eh, uhm, listening end, after the source has been filtered and attenuated to some degree, rather than in the little room. If that's an option at all. In other words, less total volume from the masking device.
Uhm.