My opponent uses Imperial Decree which resolves, then moves to my battlefield where Akali Silent sits with Guardian Angel attached, opens showdown with Bellows breath x2 which I have no reaction to so we move to resolve. How would this resolve?

Yes — Akali dies. Guardian Angel only saves her from the first kill trigger; the second Imperial Decree trigger will find and kill her even after she's recalled to base.

Sequence: Bellows Breath (with Repeat) resolves as two separate 1-damage instances, and each instance triggers Imperial Decree's "When any unit takes damage this turn, kill it" — so two kill triggers go on the chain. The first trigger resolves: Guardian Angel's replacement effect kicks in, destroying itself, and Akali is healed, exhausted, and recalled to base. The second trigger then resolves and kills her at base, since Imperial Decree's trigger tracks the unit regardless of location (FAQ #8761/#9419).

Two notes: Akali's "can't be chosen unless I'm in combat" doesn't save her — during the Combat Showdown she has the Defender designation and is in combat, so she's a legal target for Bellows Breath, and Imperial Decree's kill trigger doesn't "choose" her anyway. Also, if "x2" meant two separate Bellows Breath casts rather than Repeat, the result is the same: GA is consumed on the first kill, and the second Bellows Breath's damage triggers a fresh kill.