It's a curious idiom, and it's a real idiom that is often used. I think the explanation is that a "month" is about thirty days, so "a month of Sundays" would mean thirty Sundays = thirty weeks.
But "not in a month of Sundays" doesn't mean "thirty weeks," it means "not ever." It means the same thing as another idiomatic phrase, "not in a million years." It is impossible. No matter how much time he was given, he can't finish it.
The sentence does not hint at a reason.
It could be that the assignment is just beyond his abilities. That's likely.
It could mean someone has unfairly given him far too much work. That's possible.
The word "cannot" means he is not able to do it. So it does not suggest laziness. A lazy person is able to do something, they just refuse to do it.
By "protracted," do you mean "slow?" That isn't likely, because the expression "not in a month of Sundays" means "not ever."