Just hired a live-in. How do I determine how many hours he works each day?

Before hiring a live-in Caregiver, you need to have a plan in place so that the Caregiver can agree to do that type of work. Hiring now, and planning later could mean the Caregiver could say "no" and leave.

Sounds like the caregiver would need to be fully awake all night, thus that could be 6 to 8 hours, depending on how many hours your Dad is in his room and how often he wakes up. It would be impossible for the caregiver to sleep, and if the caregiver is a very sound sleeper, he might not hear your Dad moving about.

That doesn't leave the caregiver any time to do anything else around the house, like help you out during the day, as the caregiver would need to block off at least 8 hours for himself to get his own sleep during the day. If you want the caregiver to help you, then the time goes over the 6 hours, and will you be paying time & half for anything over 8 hours?

Sounds like you really need to have two Caregivers. One to be there at night, and another for during the day so that you can get out and about.

There will be a labour code for your jurisdiction that covers live in caregivers. It will set out minimum pay, maximum hours that he can work and time off.
With a live in you have to realize that even when they are not actively engaged in caring for the individual they still get paid as long as they are not away on personal business. Generally you would contract a set amount on a weekly or monthly basis, so counting the actual hours worked shouldn't be necessary and could cause disputes and unhappy employer/employee relations.