If Ruined Rex dies in the middle of a showdown, can I kill an 8 might unit with the trigger?

No, you cannot kill an 8 Might unit with Ruined Rex's Deathknell trigger.

According to the card's data, Ruined Rex's Deathknell ability deals 4 damage to an enemy unit. Since an 8 Might unit requires at least 8 damage to be killed, the 4 damage from the Deathknell trigger is insufficient to kill it.

Additionally, as established in RiftJudge FAQ #10090, #10182, and #10410, there is a specific timing sequence in a combat showdown that prevents this damage from being combined with combat damage to kill a unit:

1. Combat Damage Step: Units deal combat damage to each other. Ruined Rex (if lethal damage is assigned) is marked with damage and dies.
2. Triggering Deathknell: As Ruined Rex dies, his Deathknell trigger is added to the chain as a pending item.
3. Killing Units: Ruined Rex is moved to the trash.
4. Healing: During the combat cleanup, all surviving units on the battlefield are healed (Rule 461.1.a.1). Any damage they received during the Combat Damage Step is cleared.
5. Resolution: Only after the healing step is complete does the game resolve the pending Deathknell trigger, dealing 4 damage to your chosen target.

Because the unit you are targeting is healed during step 4, the combat damage it previously received is removed before Ruined Rex's 4 damage is applied in step 5. Therefore, the 8 Might unit will only ever be under the influence of the 4 damage from the Deathknell trigger, which is not enough to kill it.