It's so weird.
Sleepiness is the feeling you experience right before you actually fall asleep; the overwhelming relaxation that drags you into the unconscious. Tiredness is being too exhausted to feel or express emotion. You're awake, but mentally absent, and incapable of completing any productive task. It's one of the most frustrating feelings in the world to be unable to do something as simple as sleep. I'm extremely jealous of the people, like my roommate in college, who can seemingly will themselves into sleep within 5 minutes of laying their head down.
My insomnia was especially difficult to deal with while I was in school because it made me prone to stupid mistakes on exams. I would often misread or glaze over important details, especially on any trick or multiple choice questions, and in more than one class those little mistakes cost me grade points. It also often didn't matter how much I would study as my memory retention was horrible. Anyone who's read a book whilst tired can attest how difficult it is to not only process the words on the page, but to remember them for later.
Sounds like your entire life looks like those 2 weeks.
What kind of task can you perform that is not immediately 10 times harder under those conditions ?
P.S: I'm like your friend, I sleep like a baby. Actually less than 7 hours of sleeps makes me feel like crap.
Athletically I was able to compete with my track and soccer teammates without issue, but lifting weights is an area I lack in because sleep is so important to the development of raw strength. It isn't something I dwell on---I've lived for it so long I've gotten over it--but I do sometimes wonder in a curious way how much more I could have achieved without these sleeping issues. I used to want to be a doctor, but with this condition achieving the grades necessary to get into medical school was impossible (on a consistent basis, anyway).
Do you drink caffeine or take any other stimulants?
They could definitely have an adverse effect on your sleep.
Besides that, my impression was that injuries heal slower with polyphasic sleep.
My chronic insomnia is "primary" meaning it persists in the absence of other factors. I'm not depressed. I get enough sun exposure. I take my supplements. I have a comfy mattress. I practice good sleep hygiene. I use blue light filters and don't use my phone/computer before bed. I exercise every day. I eat well and not before bed. I don't have breathing issues. Nothing helps. It's just the hand I was dealt genetically.
Furthermore, exercising to the point of exhaustion is reckless and I've injured myself over the years doing exactly that because I don't get enough sleep for my body to recover properly.
I can run and stay aerobically fit and I played high level soccer for many years when I was younger without too much issue, but activities like weightlifting are a struggle for me. Benchlifting when you're sleep deprived is pointless so I have to fill in on my off days.
Meditation is more for calming the subconcious, dropping all the background threads that are running.
Maybe that's a "hack" that can fix your problem. Even if it doesn't, meditating a few hours at night gives your body more rest than rolling around and not getting rest nor sleep.
Maybe 4h of meditation at night is like 3h of real sleep for you, who knows.
Furthermore, you get a lot of insights and a better connection to your body.
I can recommend the app headspace.
Any "true" sleeping pill has to be tapered, so at best you can only take them for a few months at a time before they become ineffective due to tolerance. In my experience, they're also inconsistent and I would still experience a sleepness night about once a week. These are in addition to the usual side effects (oversleeping, daytime grogginess, memory issues, mood issues, etc.) that make long-term use impractical.
Sleeping pills gave me the feeling of being notionally asleep but actually conscious and awake underneath. Lot of REM sleep but hardly any of the non REM variety. That's why I gave them up very quickly. They only made a bad situation worse for me.
I'm not going to suggest anything. I feel bad for you, and just wanted to say so. I'm fairly convinced I would be either homeless or dead if I couldn't get enough sleep, and the fact that you at least aren't the latter is inspiring to me.
Your life seems impossibly difficult to me, and so I wish you all the best success in regularly finding a tranquil oasis to bide your time in along your way.
I'm sorry you got dealt that hand, it sounds like you're dealing with it far better than the vast majority of people would - including me.