Does the Sai Satcharita mention if Udi can help with sleep problems like insomnia?

Yes, Chapter 33 provides two specific examples of Udi curing sleep-related issues. In a footnote, R. Hari Sitaram Dixit recalls a friend who had been unable to sleep for eight days and found no relief from a doctor's medicine. After taking Baba's Udi for just one night, he slept very well. The chapter also mentions a friend's child from Bandra who had not been able to sleep for two months. The father took the child to Shirdi for Baba's darshan, and that very night, the child slept peacefully, and the problem was resolved from then on.