Out of the entire Bat Family, Barbara was probably the most independent. The rest of her colleagues needed Bruce, they needed that source of help and family. Barbara had her own family, her father was the Police Commissioner for Gotham. She fought because the city needed her and because it was the right thing to do.
"Sure, I'll let you know when I get tired and you can have a turn," Barbara nodded. She wouldn't tell him, she could do it on her own, but she appreciated the offer.
She accepted the key he offered and her demeanor brightened instantly. "Thanks! I'll make sure not to mess anything up." She would clean and make sure it was the way it had been when she got there. It was only fair.