Enjoy!
Sword slashes the eight desolations: Literally she says: 'sword slash eight desolations/wastelands'. Eight wastelands is an older term for 'distant places in all directions'. Her move is basically an AOE.
Three thousand strands of essence soars into battle, a sword in the sleeve follows the flying immortal: What she says doesn't seem to be an idiom. I think she's gone into shock and awe resulting in impromptu poetry. The only thing I'll point out is that while one might think the 'three thousand strands' part seems to refer to her sword slashes/powers, one of the characters can mean 'fine long hair' so it might also be referring to A-Ting's hair. I'll leave the chinese here if anyone wants to take a crack at it: 湛毫三千气冲斗, 袖中一剑随飞仙
Flying that kite: This is informing us that they deliberately triggered something to get someone to come. Hence 'flying a kite' to attract attention.
Sitting here waiting for our lead: He actually uses the chinese idiom '守株待兔' here. It refers to the story where a farmer was resting under a tree where a rabbit randomly ran into the tree and died. The farmer was happy and hence rested under the tree for longer waiting for more rabbits which didn't happen. He's saying their wait wasn't likely to succeed which is why he also says their success in baiting someone out was quite lucky.