I learned that Medicare DOES cover some nursing home care. But there are limits. After a 3-day inpatient hospital stay for a related illness or injury, Medicare pays 100% of the first 20 days if it's medically necessary for member to have a "skilled level of care" and days 21–100 are partially-covered at a lower rate. However most patients require only a "custodial" level of care (help with bathing or dressing) which Medicare does not cover. Details here: http://www.medicare.gov/coverage/nursing-home-care.html

And yes, some of the nicer nursing homes (SNF) do cost over $12,000 per month.