Thanks for the advice.
My takeaways here are:
1) I should pay her a fair wage rather than accept her free offer to watch the kids. She can then use this money to pay the expenses that we were going to pay, and we're back where we started.
2) Because I am paying child care expenses that qualifies as dependent care (unlike her plane tickets, food, and use of our apartment), I am eligible for the credit.
3) Alternatively, if I money available in my Dependent Care Savings account, I can use those post-tax-deferred funds to pay her.
4) We're talking about 4-6 weeks, she's single over 65, so she is under the threshold is well under $12k for withholding, and she makes less than $13,600 minimum filing income level even adding her home country income, so she need not file a tax return.
5) Since she is a citizen of another country, I will have to get her a work permit costing me $500. Since that is probably what the credit will be worth to me, it's not worth the trouble unless the work permit lasts at least two years and I'm sure that we'll do this at least twice in that time.
Am I taking away any conclusions you'd disagree with?