When I wake up early...

Just waking up early is already a perfect score, isn't it? It's more important to rest now instead of forcing yourself to move.

Strength training will blow away your sleepiness and improve your basic stamina. I recommend running to a nearby park, doing 10 pull-ups twice, coming home, drinking some water, and then repeating that until you hit your limit.

I have a mental illness but not much stamina, so I do light strength training a few times a week, just enough to break a sweat. I'm trying to make an effort towards something, even with 0-3 hours of sleep almost every day, without overdoing it. Setting a goal like 'I want to become like this' might be good. Also, I try to go for walks specifically during the winter each year. My knees have been bad since long ago, so I do it to prevent them from worsening even a little. I recommend doing exercises like sit-ups during the ads for point-earning games. That's what I do.