Your daughter usually lived with her mother, and unless there is solid evidence that your daughter has PERMANANTLY moved away from her mother's house, she is considered as living with her mother. That gives her mother the right to claim her as a dependent (assuming your daughter did not pay for over 1/2 of her own support).
Your daughter was over 18 for over 1/2 of the year. That means in MOST states she is 'emancipated' from her parents, which means there is NOT an option for her mother to allow you to claim your daughter as a dependent (assuming your state law is age 18). That means you can NOT claim your daughter as a dependent, which means you can NOT claim any college related credits on your tax return for her. You can NOT claim anything in regards to your daughter on your tax return.
If the mother is claiming her as a dependent, the mother can claim the college credits on her return, including the amount based on the $12,000 that you paid. If the $12,000 that you paid increased the credit that the mother receives, you may try asking her to share part of the refund that she receives. But she is under no obligation to do so.
