I want to push my daughter to be less shy
How do you plan on making her less shy?
I'm going to force her to take dance and singing lessons, and to be more active.
It is hard to see your child struggle, but you cannot push them to be someone they are not. It will not end well for her, and will cause a lot of resentment.
What am I supposed to do? I have already invested time in it.
The time that you have already invested appears to have been based on good intentions, but we need to accept that it has not worked out. Putting any more time and effort into forcing your daughter into activities that she is not comfortable with will not get that time back, it will just make the problem worse. Find out what interests your daughter, and help her to make friends with it.