Two conditions have to be met before you can make this miracle happen. First of all, your child's academic record has to be virtually flawless. Secondly, the school has to have a place for him. Let's examine both conditions a bit more closely.
How flawless does your child's academic record have to be? 99% pure is good. Evidence of solid achievement in demanding core courses shows the admissions staff that your child is a serious student. It also means that it will take a minimum amount of time to bring him up to speed in his new surroundings. The school will also look closely at how involved you are with your child's education. If they can see clear evidence of a partnership between home and school, then they will be more inclined to admit your child in mid-year.
The other consideration is the availability of a place. Nowadays families come and go, so the chances that any school has a place vary accordingly. It never hurts to ask. The other unspoken reality is that most private schools will at least consider an exceptional candidate, perhaps even placing her on the waiting list for next fall.
Can you get financial aid in these circumstances? It's not likely to happen. But always ask.
One final consideration: this is definitely a situation where the advice and counsel of a professional education consultant is warranted. He will save time and energy because he knows who to call and what to say.