I heard the Udi can help with sleep problems. Is there a story in the texts that supports this?

Yes, Chapter 35 recounts a specific instance of the Udi's power to cure sleep disturbances. A gentleman from Bandra was tormented by nightly dreams in which his deceased father would wake him with curses and abuse, causing him great distress. On the advice of a Sai devotee, he began using the sacred ash, or Udi. He was instructed to apply a little to his head before sleeping, keep a packet under his pillow, and hold faith in Shri Sai. As soon as he did this, the bad dreams ceased completely, and he was able to enjoy deep, peaceful sleep, demonstrating the Udi's natural tendency to remove such troubles.