Can I still deduct my wife this year if she is in a nursing home?

A spouse is never a dependent.  You should be filing as Married Filing Jointly even if one spouse has little or no income.  The spouse does not to be living in the home for a filing status of Married Filing Jointly.

We pay $643.43 out of her SS retirement check to the nursing home each month. Can we take that off our taxes this year?

If you, your spouse, or your dependent is in a nursing home primarily for medical care, then the entire nursing home cost (including meals and lodging) is deductible as a medical expense.

IRS Publication 502 - https://www.irs.gov/forms-pubs/about-publication-502