You can claim dependent care expenses regardless if she files or not.
However, not having her SSN complicates matters. You must have that information in order to file. There is one exception when you don't have to have it - it's when your cousin refuses to give you her SSN.
According to the IRS Publication 503,
page 8, Provider Refusal, “If the provider refuses to give you the
identifying information, you should report whatever information you have (such
as the name and address) on the form you use to claim the credit. Enter
"See Attached Statement" in the columns calling for the information
you do not have. Please attach a statement explaining that you have requested
the information from the care provider, but the provider refused to furnish
that information. Be sure to write your name and social security number on this
statement. The statement will show that you used due diligence in trying to
furnish the necessary information. " You will have to print and file
the return by mail.
