I.G.
Hitting and biting is a very normal phase at that age.
While there certainly are different ways to deal with unwanted behaviors, I would suggest that in this case to you refrain from a "pat on the butt" - kinda hard for a child to understand that hitting is not okay, except when mommy and daddy do it Edit: Sorry I read taping not tapping. I still think though, that it is counterproductive, if you are trying to teach that hitting is not ok.
Whatever you decide on for discipline you have to remind yourself of two things: there are no quick fixes and you must be consistent!
Just because you tried time outs yesterday, it does not mean that she will not test her limits again today or an hour later or five minutes later for that matter. At this age they are still learning the concept of action and response, so your response must be predictable all the time. Pick something and stick with it.
We went with a warning/timeout method. First time hitting I issued a warning, a reminder that hitting hurts and it is not nice and if you hit, you will sit in time out. Next time hitting, I put her in time out. She usually stayed there no more than a minute. We also taught her to apologize and give a gentle touch. It took a few weeks, but the hitting and biting have since completely stopped.
For me this age was very difficult, I had trouble adjusting from parenting a baby to parenting a toddler, but as much as she seems, grown up and cocky, laughing at you when you discipline her, remind yourself that she is STILL a baby and that she will learn to respond to gentle discipline just like she would learn to respond to physical pain or restraints. It may take a little longer to learn but is very much worth it.