How fast the brain recover itself at sleep? What can be done to accelerate this process?
You should see some talks from TED about deep sleep. 

One researcher summarized that adults generally need 8 hours every night to be at optimum strength. What is most important though is the Delta-wave sleep that occurs during very deep sleep.  To accelerate the process: It's pretty fringe research, what he said is early days... Alledgedly, playing a certain kind of noise triggered near the delta wave phase promotes Delta wave activity in the brain as measured with electrodes. it's essentially slow delta-synchronised seashore kinds of of grey noise played in sleep at the right moment. He said they were working on a device for it. 



here's the talk. 
https://www.ted.com/talks/dan_gartenberg_the_brain_benefits_of_deep_sleep_and_how_to_get_more_of_it

My own method to have amazing sleep is lots of nights at my girlfriends and to change mattress every few days mat my house, because my unconscious breathing is forced to exercise different muscles of the chest based on whether I am on a swimming pool matress, a futon, a cozy matress, covers similar to sleeping on a lawn. Physiology and muscles benefit from varied exercise, including the 8 hours of chest muscles during sleep. another matress for example I feel all the breathing in the lower belly. I just got out a pool matress for the last 2 nights on boards and rugs, and I dreamed that I was flying around the mountains and cities for about 3 hours and that other people were flying too, which is always nice. I swear I was sleeping a bit rough 3 days earlier on a normal mattres. other factor for me optimum warmth and lots of water for me:) i'm weird though because I had liver damage from a pesticide and following doctor care when I was 18 and I need drink more and sleep hot in order to sleep well.