For the past eleven years I have known clearly what I was meant to do, but the time was never right. While I was growing up, my family ran a homeless and battered women's shelter.
These two sentences do not go together. The thing to do is write a good intro sentence and then follow it up with something that builds on it. I like your opening sentence, but please elaborate on it in the second sentence.
Oh, ha ha, okay, now that I actually read the 3rd sentence, I see that the only thing necessary to fix it is to put it all together as one long sentence:
While I was growing up, my family ran a homeless and battered women's shelter, and every day brought someone new with some issue -- usually legal -- that kept them in a stagnant and unproductive lifestyle.
By doing this, you make it sound like it makes sense. But without putting those 2 sentences together it creates a moment of confusion for the reader at the end of that second sentence, and it is no good to confuse the reader right at the start of the thing...
Every day must be 2 words here in the first para. As one word, it is an adjective.
I knew this is what I wanted to do from a very young age. Now, this sentence ends the first para, and I think it is not good enough. You already said you knew this from a young age. Use the last sentence to help the reader prepare for what is coming, as a way of giving the reader a sense of what is going on so she does not have to follow you from one idea to the next. You can write a sentence that says, "This essay is intended to explain... and also to give some of my ideas for wanting to ...
And that way, the thesis sentence will "support" all these ideas, including the one about your grandfather, which is a perfectly valid and very nice reason for wanting to succeed in law school.
:-)