It really depends on the material; are these sounds that you want to remove truly peaking at a lower level than every bit of the speech? How about in loudness or 'energy'?
You need to look into the kind of sounds you want to remove and see how they differ from the ones you want to keep. This is key to being able to separate them.
Are they of a shorter duration than the sounds you want to keep? Are they at a different frequency? Do they have some sort of regular pattern?
I'm just trying to get you to think in the right way. You need to first define a difference before you can look for a tool to remove them.
You have established that the unwanted sounds are at a lower amplitude than the sounds you want to keep. This makes things very easy; all you have to do is find a tool that will allow a signal to pass through(or open the gate) if the amplitude passes a certain threshold, and reject any signals that fall short of this threshold(close the gate). It's almost as simple as that.
I say almost, because there are other parameters to consider. How fast do we want the gate to open and close? because we don't want any sharp clicks and we may want a smooth transition through the attack and decay of this opening and closing gate. Also, do we want the closed gate to completely attenuate the signal, infinitely, or do we want to just gently attenuate the signal by, say 12 dB? Do we want the left and right channels to be 'gated' individually, or do we want to treat them as one signal? These are the basic extras that a generic gate provides.
If you are having trouble with Audacity's Nyquist gate then I would suggest you look for a free VST gate online. There are many to choose from, pretty reputable companies release free VSTs if they are simple enough, like noise gates.
best free noise gate vstau plugins from bedroomproducersblog.com
unfiltered audio g8 cm free vst au noise-gate plugin from musicradar.com
Other cool free VST FX(Voxengo)
Immense list - free Gates/DeEssers
I think your main problem is understanding how to use these gates or de-essers(de-essers are just gates that target high frequency sounds like "sssss" sounds (or "sibilant frequencies"))
I explained the basic parameters of a gate, but there is one more that could be helpful to you, it is an internal sidechain feature, sometimes called other things but basically, it allows you, like the de-esser, to target certain frequencies with the gate. It usually consists of two parameters, an upper and a lower cutoff frequency. So if you set the lower cutoff to 2 kHz and the upper to 5 kHz, then the gate will only be triggered if the amplitude of this range(2-5 kHz) passes the threshold. So these inhalation sounds could effectively be targeted in this way.
So a word on setting up a basic gate;
To initially set up a gate, you want to set the attack/decay to a low level; zero if you like, set the attenuation(level reduction) as low as possible, then sweep the threshold down, starting at 0 dB. This will give you a good idea of when the gate triggers, so you can then set the threshold to the appropriate level(just above the unwanted noise), then adjust the other settings to fine tune.