Can my wife and I get reimbursed to care for my father with dementia who has had a stroke?

Be careful. If you have POA you can reimburse yourself for expenses incurred on his behalf. You cannot pay yourself for your time. If he has to go into a nursing home you will probably need medicaid funds since the nursing home will cost ~$300 per day. When you apply for medicaid the state will ask for bank statements looking back 5 years. If they find irregularities you could have to pay back anything they don't think you were entitled to.