If I attach Svellsonger onto Akali Deadly Weapon, will she deal 4 damage when she moves into a battlefield, killing a unit with 4 might?

Yes. If Akali is Empowered, attaching Svellsongur gives her a second independent copy of her "When I move" trigger. Each trigger deals 2 damage (instead of 1), for a total of 4 damage to a unit at a battlefield she moved to or from — enough to kill a 4‑Might unit.

If Akali is not Empowered, each trigger only deals 1 damage, totalling 2 — not enough for a 4‑Might unit.

Both triggers can target the same unit, and since the ability says "you may deal... to a unit" (not "another unit"), they can both hit the same target without issue.