How do we go about getting Dad to a nursing home? Do we need a doctor or hospital?

Will Medicaid be involved, or will he be self-paying?

Do your mom and dad get along well? Has he been visiting her often? (Is the staff there familiar with him?)

Talk to his MD and the nursing home where mom is. Hopefully they can accommodate him, see whether they keep couples together or in nearby rooms. It might actually take 6 months to get an opening. Once you have an admittance date, give the landlord notice. If the apartment can be rented quickly, he will let you off the lease. Also read the lease to see if it renews automatically and how much notice is required.

Call the facility that your mother is in and see what they can do to accommodate your father.

Is your concern about breaking the lease? I'd call his landlord/landlady and explain that your father needs to go into a nursing home. Basically see if they'll let him out of the lease because of that. Maybe offer an extra month's rent, so the landlord can re-rent his place without losing money. Or see if you can sublet it. But I'd try to get him out of the lease by explaining the situation.

And then put him in the facility where your mom is.