I've started (as of last Thursday, I think) taking melatonin. I originally took 1mg a night. I did not notice any effect from this. For the past few nights I've taken 2mg. I'd expected that this would cause me to feel tired half an hour after taking it, but this doesn't seem to be the case. But I have been going to bed anyway about half an hour after taking it, then reading for a bit, and then successfully sleeping for about eight hours. (Previously I would often lie awake for what-felt-like ages; sometimes literally more than an hour.) And my sleep cycle hasn't precessed this week, which it did the last few times I had no schedule to wake up for. So it seems to be an improvement. I haven't noticed myself feeling any different throughout the day.

Further experimentation: I should try 3mg, and I should try skipping some days. I'd also like to bring my sleep cycle back by a few hours by Saturday night, because I need to get up early on Sunday.