Heather, How is the new student to work with? Has the mother indicated why so many schools? Maybe it's not so much the boy, but the mother is looking for something and doesn't know what it is she wants for her son. A child that has bounced around a number of schools before the age of 8 sends up the red flag to me.
Here's the thing, she was aware of what you offered in the way of homeschooling, you both have spoken previously and had it nailed down, so why do you have to change the way that you homeschool to accommodate this one child? She knew going in how you taught your classes.
Worksheets are the norm and they do help the children apply their thinking and can actually see what they've done. Worksheets are used in the public and private schools and yes, even in the grown up world of business. It's nice to have some hands on time as well as work sheets, but the expectation of hands on and unschooling for most of his subjects doesn't sound like he'll benefit much from it w/o the actual thinking part.
I know you would do a good job schooling this young man, but are you ready to change your way of teaching for one child? Wouldn't this disrupt your other students' school time?
Something just doesn't sound right to me about this mother and the number of schools this child has attended. If the child is happy and he has had several good days, why not wait and see how the child does for several weeks? Two days isn't enough time to determine if your homeschooling is or isn't the type of schooling he needs.
Just my two cents.
Sit quietly, the answers will reveal themselves when you least expect them to. The past is gone, the present is a gift and you need to focus on today, allow the future to reveal itself when it is ready.