I have also had great results with meditation aiding my sleep. For me it was mindfulness meditation, and recognizing when my mind was buzzing or going in circles, then letting the thoughts go and coming back to my breath or just observing my empty mind and watching for the first sign of a thought.

For years I was hugely skeptical that saying words in my mind would make a difference, but when I tried a sleep meditation, like a yoga nidra, mentally saying things like “i have nothing else to do, and nowhere else to be. i am here to relax.” really helped. also, when my mind starts going off on a crazy line of thought and i recognize it, i can usually get it to stop by mentally saying “i can think about this tomorrow.” i’m still amazed that it even works. the idea that you can so easily trick (or i guess vocally control?) your own mind just sounds ridiculous. I suppose though that different minds work differently, and the mental words may not work for everybody, or not all the time.