Most of the special needs jobs that list $10-15 are set by government reimbursement or payroll. I've asked for more information on two and both were $11.47/hour. There are numerous families with children that have special needs for which the parents are willing to double that salary, though the extra is typically off the books.

$20/hr.

$25 seems right, if you are even willing to take it on at all. Such a big responsibility.
The mom has to know that nannies get paid $10-$15 for 1 child, no special needs. Why would a nanny take on so much extra work for no extra pay?

She really needs a nurse. At least $25, but she may be getting a gov't grant or insurance and it limits how much can be paid.

$50/he. Too much work.

Absolutely not $10-$15. I would charge at least $18 to $25 an hour and I don't have the qualifications or experience to handle a special needs child.

I received a message from a mother seeking care for her 11 year old special needs child. The child has a heart defect and a colostomy bag, along with cognitive delays. She said she wanted to set up an interview with me because she thinks I have the qualifications she's looking for - I do, and could certainly handle the situation. She listed her pay scale as $10-$15 an hour. There's no way I'd accept that pay for the level of care her child will require and I let her know that. What would you charge per hour?

I don't live in DC, but another large metropolitan city.