Complex sentence: At the same time, the Black Hawk War had recently begun in Illinois, and the battle of Stillman's Run, fought on May 14, proved to be a victory for Black Hawk's band over the Illinois Militia.
Split into simpler sentences: At the same time, the Black Hawk War had recently begun in Illinois. The Battle of Stillman's Run, fought on May 14, proved to be a victory for Black Hawk's band over the Illinois Militia.