Which kingdom's army fought William's Normans at Hastings?

The army that fought William’s Normans at Hastings belonged to the Kingdom of England. It was led by Harold Godwinson, the Anglo-Saxon king crowned after Edward the Confessor’s death in January 1066.

Harold’s forces were not a modern national army. They included housecarls, heavily equipped professional or household troops, and the fyrd, a broader levy of local fighting men. Most fought on foot, forming a shield wall on Senlac Hill. The English army had recently marched south after defeating the Norwegian invasion at Stamford Bridge, leaving it tired and partly depleted.

William’s army was drawn from Normandy and other parts of northern France, with contingents including Bretons and Flemings. Its combination of infantry, archers and cavalry eventually overcame the English defensive position. The English defeat killed Harold and several prominent members of his family, including his brothers Gyrth and Leofwine.

A common mix-up is to identify the English defenders as “Anglo-Saxons” rather than soldiers of the Kingdom of England. “Anglo-Saxon” describes the culture and ruling elite; the kingdom was England, and that is the canonical answer here.