One solution would be to hike faster which would cause you to breath heavier and deeper. Most would probably not advise that.
Agree. The best correlate with development of AMS is rate of ascent. Those who go faster are at greater risk.
You may want to look up "pressure breathing" a mountaineer technique.
However, I'm not sure I buy that "not breathing deeply enough" had anything to do with it.
Good quality sleep prior to the hike may or may not have had to do with the AMS, but it sure didn't help.