There is a simple breathing exercise for insomnia developed by Russian doctors practicing the Buteyko holistic self-oxygenation therapy. The exercise was used by many thousands of people in Russia and taught to them by over 200 doctors. The exercise helps to fall asleep much sooner. (I translated many Russian medical papers.)
In order to be successful, the following are preliminary requirements.
1. Go to sleep when you are really sleepy.
2. Before and during sleep breathe only through the nose.
3. Do not sleep on your back.
 
Steps to follow
Relax all your muscles when you are lying in bed on your tummy or left side. Focus on your breathing for a minute. What do you feel? If the sensations are vague, take a deep but slow in-breath and relax to slowly exhale. Do you feel the airflow going through your nostrils? Do you have any sensations at the back of your throat? Are there any feelings about movement of air inside the chest and bronchi? What do you sense near your stomach?
Next, instead of taking your usual inhalation, take a slightly smaller inhalation (only about 5-10% less) and then immediately relax all muscles, especially upper chest and all other breathing muscles. Take another (smaller) inhalation and again completely relax.
With each breath, take a small or reduced inhalation and then completely relax. You will soon experience light air hunger. The goal is to preserve this light comfortable level of air hunger for 2-3 minutes.
The breathing can be frequent during this reduced or shallow breathing but this is OK. If you do the exercise correctly, you will notice the following signs:
- The arms and feet will get warm in about 2-3 minutes after starting the reduced breathing (this is the central sign);
- The nasal passages will become moist and the nose colder in about the same 2-3 minutes.

A more advanced version
A more advanced version of this exercise involves stress-free comfortable breath-holds for faster results. However, people who suffer from heart problems, hypertension, panic attacks, or migraine headaches should avoid any breath holds. These people can stick with the above-described version. How is this version with breath holds done?
After your usual exhalation, pinch the nose and hold your breath only until the first discomfort. Immediately release your nose, but instead of your usual inhalation, take a small inhale, and then relax for exhalation. Continue the reduced breathing as it is described above.

Conclusions
Our breathing pattern has profound effects on sleep. For example, breathing through the mouth affects many biochemical and physiological processes in the human body. Sleeping on the back can make breathing about 2 times heavier thus reducing oxygenation and triggering sleep apnea, mouth breathing, headaches, anxiety, panic attacks, and many other problems.
If you retrain your breathing pattern so that after your exhalation you can comfortably hold your breath for 40 or more seconds, your sleeping problems will disappear.