Thanks to everyone who left recommendations for sleep aids in the comments.
At the suggestion of a pharmacist, Chava got me an Israeli product called "Rigeeyon lie-la" (night relaxation), whose ingredients include "concentrated standardized extracts of passion flower, skullcap [I assume they do not mean yarmulkes!], hops, and lemon balm."
I went to a wedding last night and got home around 11:30 or midnight. Took the pills around 12:30 and lit a scented candle (I know you aren't supposed to leave on a candle while sleeping, but I do it with the Shabbat candles all the time, so why not this? Besides, I put it on the floor in the middle of ceramic tiles with nothing around it.) and tried to go to sleep. After about an hour I fell into a fitful sleep, woke up at 4 AM, blew out the candle because I realized that I was, deep down, worried about burning to death in my sleep, and after that I slept well until about 11 am, when Chava called to find out if the pills had worked. :-)
On Tuesday I have a doctor appointment to deal with my hand; I'll ask her about serious, serious sleeping pills as well.
Thanks for all the support!