The site continues to tell us that my wife is eligible for a tuition deduction, but it's not being added to our refund or accessible in any way. How can we claim it?

The Tuition and Fees Deduction (TFD) is only a reduction in income before tax is calculated.  It will show on line 21 of Schedule 1 and line 10a of form 1040 (along with any other schedule 1 adjustments/deductions). 

If, instead, you are asking about a Tuition Credit, it will show on form 8863.  It may or may not include a refundable portion.  That is, most or all of the credit can only be used to reduce you tax. It is not necessarily added to you refund. 

Before concluding that TurboTax is doing it incorrectly, check the actual forms,  Don't just go by the "refund meter" at the top of the screen.

Just because you are eligible to take the credit, that doesn't mean that you will benefit from it. If your tax liability isn't high enough and you don't qualify for AOC, then it may not help you.

The best thing to do, however, is to go back over all your answers and make sure you answered correctly.