Frustrated by existing sleep-time applications, I wrote a simple application to play music on an Android phone. The application is called Sound Sleep and it is available for free on the Play Store.
To use the application, create a subdirectory called "sleeping" under the "music" directory in the SD-card or external storage. Put night-time sleep music in this subdirectory. Many music file formats (mp3, ogg, ...) are supported. All directory names are case insensitive.
Start the application. The top half of the screen starts and stops night-time music.
The bottom half starts/stops white noise.
Since the application allows tapping anywhere, it is perfect for clumsy night-time use. It works on all Android devices starting with version 1.5, so give it a try!