Talk to Hamanasi. They have their own in-house dive shop on the property, and their boats are right behind the resort.
The diving should be perfect for a new diver as long as the water's smooth. Typically the first dive will be to 70 feet max and the second (and third if they do three) to 60 max with most of the dive around 50 to 60 . The water will be warm, and the viz will be great. Usually no current to speak of. I'm sure they offer nitrox. Not really "necessary" but I prefer it since I pack in a lot of dives when I go to Belize.
As far as the Blue Hole, I don't know if they run trips that far as that would be an all day affair. If they don't, you could contact Splash in Placencia to see if they're going while you're there. They do the BH if they have a certain minimum number of divers, I think six.
I've never stayed at Hamanasi, but I've had lunch there and toured the property. It's very nice. I'm sure you both will have a great time. Be sure to let them know your wife is new to this. They'll take good care of her. Hamanasi has its act together.
Make sure you pop over to Chef Rob's restaurant at least once.
Look forward to your trip report.