Sympathies. I cannot abide intrusive noise, having always lived in very peaceful environments, and am child free, so find it hard to tune out any external distractions, as I have never had to.
I live next door to a teenager with severe autism and other complex issues who has frequent meltdowns and outbursts of stamping, screaming and bellowing.
I looked into sound proofing but remained unconvinced that it would completely solve the issue. Very expensive too.
Bose noise cancelling headphones are fantastic and worth every penny. Mine don't play white noise automatically but the sound quality is fabulous and it cuts out the vast majority of the noise except the very highest pitches.I would highly recommend. But understand you don't necessarily want to wear headphones all day.
I re configured my house too, so I am not spending all my time in the room where she is most active. Though I appreciate that is not probably possible for you.
I have found that as soon as it starts up, if I put Radio 3 on immediately, although I can still hear the noise, I fixate on it less, and I find the music soothing.If it's insupportable I go out for a walk, or to the gym, or do some other distracting thing.
I have noticed it is affecting me less than it used to do, which is miraculous as I was considering moving as the only option at one point.
Would it be worth speaking to the mother in a nice way? You should be able to enjoy your home in peace and quiet, at least some of the time, and as others have pointed out this noisy child could well grow into a noisy teenager.