This is a perfectly valid use of Divine Smite and a smite spell
We can see in the description for Divine Smite that the attack must be a melee weapon attack:
Starting at 2nd level, when you hit a creature with a melee weapon attack, you can expend one spell slot to deal radiant damage to the target, in addition to the weapon’s damage. [...]
We don't have any additional restrictions on the use of Divine Smite, other than that we must have a spell slot to expend.
The various Smite spells available at time of writing (Banishing, Blinding, Branding, Searing, Staggering, Thunderous, Wrathful) all begin with a variant of the following:
The [next] time you hit [a creature] with a melee weapon attack...
There is not a restriction saying you cannot combine these two effects. So, given they meet all requirements (a melee weapon attack and a spell slot to expend), a paladin could indeed use both a smite spell and the Divine Smite feature on a single attack.
We can also look in the Sage Advice Compendium for further clarification:
Can my paladin use a smite spell along with Divine Smite? As in, I cast wrathful smite, hit, then use Divine Smite on the same attack?
Yes, you can use Divine Smite on the same weapon attack that benefits from a smite spell, such as wrathful smite—as long as the attack you make after casting the smite spell is a melee weapon attack. Divine Smite doesn’t work with any other kind of attack.
While this is certainly powerful, it is also resource-intensive. A low-level paladin choosing to do this might struggle to perform during the rest of the adventuring day.
The sword itself is making the attack, it's just using your modifiers
The dancing sword item description says (emphasis mine):
...the sword begins to hover, flies up to 30 feet, and attacks one creature of your choice within 5 feet of it.
While the sword hovers, you can use a bonus action to cause it to fly up to 30 feet to another spot within 30 feet of you. As part of the same bonus action, you can cause the sword to attack one creature within 5 feet of it.
Compare this to a similar effect, the spiritual weapon spell
When you cast the spell, you can make a melee spell attack against a creature within 5 feet of the weapon.
In the case of the dancing sword, no reference is made to you making an attack. The sword is always the thing that is attacking, it's just using your modifiers.
Based on this we may conclude that the sword's attacks aren't made by you. If we were meant to consider the swords attacks to originate from you, it would follow a similar wording to spiritual weapon. That is, it would say something along the lines of "you can make a melee attack against one creature..."
For that reason, effects that apply when you hit with a melee attack do not apply to the attacks made by the sword while it is hovering.
Best Answer
Adding a 1d8 radiant damage to a magic weapon is fairly straight forward:
This should make the d8 radiant damage be added as a note on the attack option.
However, this doesn't make it automatically rolled when you roll for damage (this might have something to do with dice and mixed damage types). There is a workaround for this though, but it only works when the damage type doesn't matter and you're not using the versatile option. Set up a custom attack action.
Hit manage custom, and create a new action option. You'll want to set Range to Melee, Stat to STR, and Activation type to Action and mark Proficient and Display as Attack. Then set Dice Count to 2 and Dice Type to d8. You can use Fixed Value (to the right of the dice values) to add a static bonus to the damage roll, such as from Dueling. And give it a name so you remember what the attack option is. This will at least give you a conveniently accessible 2d8 + STR damage roll.