With Soraka, if you use Blade of the Ruined King to kill something with less might than her to equip it, does that unit get healed from Soraka?

Yes — the unit gets healed, exhausted, and recalled to base by Soraka, and you still successfully equip Blade of the Ruined King.

Soraka's ability is a replacement effect (Rule 369). When you attempt to kill a friendly unit with less Might than her as the equip cost, Soraka's ability replaces that death with "heal it, exhaust it, and recall it" instead. The unit doesn't die — it gets healed, exhausted, and sent back to your base.

Crucially, per Rule 357.2.a, costs that are replaced by replacement effects are still considered paid. So even though the unit was saved, you've still fulfilled the equip cost, and Blade of the Ruined King attaches to its target normally. This is directly confirmed by FAQ #374 and #9045.