Can you recharge without sleeping?
No, you cannot. Although we don't know how sleep works, we know that lack of sleep is very devastating to the body, and nothing (including laying in bed relaxed but awake) can replace it. If a rat experiences 100% lack of sleep (somebody keeps it awake all the time), this will kill it sooner than 100% lack of food would. There are no wake-until-you-die studies in humans, but even reduced sleep has very bad effects, both cognitive (it impairs you similarly as if you were drunk) and physiological (metabolic changes: you need more energy for the same actions, and food intake increases disproportionally). 

If you have insomnia (and yours sounds severe), you need to get it treated. There are different options depending on whether an underlying cause is found, and whether behavioral therapies turn out to work for you. There are no alternatives which replace sleep. 

The above is just a summary. For a dry but very reliable source on sleep, see Principles of neural science (ed. Kandel et al), Chapter 51 written by David McCormick and Gary Westbrook. For a very good popular science book on the subject, see Sleep thieves by Stanley Coren, I can recommend it for anybody with insomnia.