My dependent is not showing up on the list of "whose care did you pay for" of the dependent care deductions page. Any help?

Have you entered any earned income from work on your 2023 tax return?

There are several other tests you must meet to claim the credit, from About Form 2441, Child and Dependent Care Expenses:
 

• You (and your spouse, if you're married) must have "earned income," meaning money earned from a job - unless you are a student or disabled. Non-work income, such as investment profits, doesn't count.
• You must have paid for the care so that you could work or look for work. Being a full-time student or a parent unable to care for themselves does count as "working" for the purposes of the credit, even if you don’t receive any income for it.
• If you are married, you must file a joint tax return.
• You must provide the name, address and Taxpayer Identification Number (TIN) of the person who provided the care. The taxpayer ID number is either a Social Security number (SSN) or an Employer Identification Number (EIN). Ask your care provider for the number.

TurboTax has an article here that may be helpful. Another tip, don't enter zero for wages, better to leave it blank if you are a student.

Reference: Claiming a dependent