"My daughter takes 10 minutes to actually say 'OK' to an instruction. How do I speed things up?"
Mom: "Nancy, go clean up your room."
Nancy: "But none of the mess in there is mine."
Mom: "We need to clean all the rooms in the house right now even if the mess isn't ours. I need you to go clean your room please."
Nancy: *pout face* "You always ask me to clean the room. Lizzy never has to help me."
Mom: "Lizzy is smaller than you. She helps you the best she can, but big people always have to do more than small people."
Nancy: *crying* "You never care how I feel! You treat me like Cinderella."
ETC. ETC. ETC. This power struggle could go on and on.
This is how I would handle this situation:
Mom: Nancy, go clean your room.
Nancy: But none of the mess in there is mine.
Mom: Nancy, just now I gave you an instruction and you looked at me, but you didn't keep a calm voice and face and you didn't say "OK". You are not following instructions. (At this point if my child was new to following instructions, I would tell them I was going to give them the instruction again so that they could follow it correctly, and then give an instruction. If my child choose not to say "OK" or disagree appropriately I would tell them that is seemed like they were out of instructional control and I would start the rule of three.)
This way does take a little more talking, but it dispells the power struggling that takes even longer than just doing the correction. It also helps the child change their heart and learn faster. To help them follow instructions faster, do corrections when they don't follow the steps.