Rumble, Hotheaded is defending a battlefield alone with +1 might from ability coming from other source and +1 shield coming from Rumble legend. Now I send Empowered Akali, Deadly Weapon dealing 2 damage from her ability when she moves, and I activate Empowered Akali legend to send her back to base, then another 2 damage because Akali moved. Since the Akali legend is empowered, Akali, Deadly Weapon is ready. Now I send her again to battlefield where Rumble, Hotheaded is there, dealing another 2 damage. Now they are there together. What will happen next?

Off on the second combat might of Akali.

Rumble, Hotheaded survives the first combat; Rumble and Akali, Deadly Weapon end up dead, and you don't conquer the battlefield. The critical point: your first two hits (2 + 2) do not stick into the second combat. Because Rogue Assassin is Action speed, you can only use it after the first Combat Showdown has opened — and an initiated Combat Showdown must clean up, so Rumble's first 4 damage is healed back to full (FAQ #11658). He enters the final fight at 0 damage. Final fight: Rumble defends with Might 6 (4 base + your +1 buff + 1 Shield from Mechanized Menace). Your final move trigger deals 2, then combat damage from 4-Might Akali adds 4 → 6 total, which is lethal against 6 Might. Meanwhile Rumble deals 6 to Akali (Might 4 as attacker, no Shield) → lethal → she's killed. Combat Cleanup ends with an empty battlefield. Note that going straight to the set up of the second combat is the prefered play sequence.