I guess the first question I'd ask is if your boss or co-workers are noticing you nodding off. I would take immediate steps to limit this if other people are noticing. Smoking isn't a very good option, because getting up and moving around is better for you. The nicoteine in cigarettes is only really effective in waking you up if you smoke two or three cigarettes, and you don't have that kind of time at work, plus it's really bad for you.
Chewing gum is not a long term solution, and eventually it will give you headaches or TMJ. Periodic jolts of caffeine help, so if you can get a cup of coffee or regular tea every hour or so, that could make a difference too. Since you are waking up so early, a good exercise workout is a great way to get you mind and body alert.
I would make an appointment to see your doctor immediately, and see if he can switch around your medications. In the meantime, if your boss and co-workers know about your problem, let them know it is due to medication (don't tell them what it is for), and that you are taking steps to change the situation.
As far as losing your concentration while reading a work text, everyone gets distracted from time to time, especially if you are bored with the subject matter. Try to schedule your reading time during the day when you are most alert if you can.
The distraction factor may come from playing the video games. They are very addictive and can be very mind-numbing. Instead, try playing word/number games like crossword puzzles or Soduku. These make you actively use your brain, rather than numbing out your thoughts, which tend to be very limited when you are playing video games, which are much more hand/eye coordination based. Also, video games are very addictive, and you can lose valuable time by spending hours and hours transfixed on the TV screen.
One last thing, try to boost your consumption of B vitamins, as these are good for energy.
Let us know how you are doing.