The visitors took an early lead through Jamie Proctor's 10th-minute opener but King scored his first goal for the club after 63 minutes to earn the Blades a point and move David Weir's side off the bottom of the table ahead of Saturday's fixtures.
Crawley then took the lead when Andy Drury's cross from the right-hand side was met with a close-range header from Proctor at the far post.
The Blades improved after the break as Jose Baxter's shot deflected wide and Tony McMahon struck the woodwork with a long-range effort.
King equalised just after the hour, slotting past Jones after a ball from Brandy found him in space inside the area.
As Weir's side pressed in the closing stages, Brandy headed wide and substitute Conor Coady's shot was turned wide by Jones.