Hello. Your just a few counties over from me-welcome to the site. Have you done a check on your hive to see if you do have a queen and of so is she laying good? Also, when you get a chance change your location in your profile-there are several Tn Keeps on here to help you out.
I did a mite count with the sticky board this past weekend and that looked very good. I was in the hive yesterday and they look well and happy. The bees were returning to the hive loaded down with goldenrod pollen.
The problem I found was that a laden 10-frame deep is HEAVY. I will be going to 8-frame mediums next year.
A 5 frame queenless nuc made in mid August shouldnt have a high mite load with the brood break. After re-reading your post, are you still feeding 3 gallons a week? I cant imagine a 5 frame queenless nuc needing that much syrup. Your hive was queenless for at least 15 days, a few days to mate, and then another 21 days for her brood to hatch. Just make sure that your hive isnt honey bound and really make sure that you have a laying queen. Meanwhile, read, read and read some more and enjoy!