Since joojaa advised to use Shape Builder in my 1st answer, I tried it and it worked as expected. The thing is, what I liked in Live Paint feature is the gap detection part. I didn't know it is present in Shape Builder too. I just checked and enabled it.
So just make sure you Check the gap detection option in case some of your paths are not closed and it works pretty well. (Adjust the gap detection length according to your requirement. In my case, gap length was not necessary as all paths were closed, so it just created unnecessary 2 shapes for eyes, which I removed later) Once you're done, remove unnecessary strokes. You don't even need to expand the object. Fill the colors individually.
Note: There are many ways you can use shape builder. You can merge, add, subtract shapes. You may want to explore more about how to use it.