Why does it have to be so hard?

Here lately Brooke has been very challenging for me, when I ask her to close a door that is open she says "but mom I didn't open it" or "it's already closed" when it in fact it isn't closed. The thing is, is that when we ask her to do something she is supposed to just do it, that is how I was raised. Yesterday for example Brooke, Kyra, Tyler and Taylor were rowdy ALL day. Kyra was coming to me and telling me Brooke hit or scratched her and Brooke vice versa. I went to talk to the girl and Brooke kept talking back to me, so as punishment I told Brooke she was to lay on the recliner in the living room with no TV, this she did not like. She kept getting up and saying she was thirsty or she had to pee (when I knew for a fact she didn't have to do either) and she wasn't minding me. So I spanked her, I explained to her that she was in trouble and she was not get up from the recliner until I told her she could. I explained to her that using the restroom or thirstiness as an excuse to get up was no longer tolerant for me and I would spank her again if she got up. Well apparently I didn't spank her hard enough the first time because she got up AGAIN!! :( Oh how be the authoritative parent is so so challenging with her. So I spanked her once more, well then I needed to shower and Taylor was napping so I "threatened" her not to get up while I was showering or she would just have to get another spanking, which I'm trying to so hard to consider another method of discipline, well by then Jason was done washing his truck so he was inside handling all the kids while I showered. What is bugging me most is I don't want my precious Brooke to think I don't love her, or to think that I favor her brothers over her. All this is, is just a milestone in our lives we are going through. Children aren't perfect, that's why they have parents to teach them right and wrong and respecting others. This is a battle I am having with her and I just pray that I am dealing with it the right way. Jason and I have to establish ourselves are the authority with our kids and unfortunately Brooke is the first born so she will experience it first. I just love her so much and I feel like I'm being so hard on her, but I can't lose this battle with her. :( It's a hard spot to be in as a mother and I just hope I'm handling this with love and patience. I keep reminding myself of that fact before I intervene whatever situation is happening at the moment. I have so many expectations of myself and through other blogs from mother's that have been or are going through this same situation I know that this is just what we have to do. I hope today ends better.