If a long term care policy won't cover all of the costs for a skilled nursing facility, will Medicare cover some of the residual amount?

Medicare won't cover it. Medicaid will after she spends all her money down to no more than $2000.

Medicare can pay all of the cost of the first 20 days in a skilled nursing facility and most of the cost for the next 80 days, if she had a 3 day prior hospital stay and her care is rehabilitative (which it is in this case). If she has a medicare supplement, then that will cover what Medicare doesn't cover for days 21 through 100. The admissions person at any skilled nursing facility should know this and should be able to confirm it for you.