Casting Lightning Crafter with only Kiki-Jiki, Mirror Breaker out can be done in a way that doesn't leave Kiki stranded in exile:

Cast Lightning Crafter (Crafter 1); it enters the battlefield, and its "you must Champion something" trigger goes on the stack. In response to that happening, use Kiki's ability, targeting Crafter 1, creating Crafter 2. Crafter 2 enters the battlefield and causes a second "you must Champion something" trigger on the stack. In response to that trigger, Crafter 2 can use his ability to deal damage to something (since it has haste). Then let the Crafter 2 "Champion" trigger resolve, and target the tapped Kiki, which moves him to exile. Let the Crafter 1 "Champion" trigger resolve, targeting Crafter 2. Crafter 2 gets exiled (and vanishes, because its a token), which causes Kiki to come back from exile (untapped!).

From here you could create a loop by using Kiki's ability on Crafter 1 again (creating Crafter 3), having Crafter 3's "Champion" ability target Kiki, then having Crafter 3 use its ability on itself, which will kill it, and bring back Kiki. This loop creates a bunch of "a creature enters the battlefield" and "a creature dies" triggers. This would be useful if Purphoros, God of the Forge, Rockslide Elemental, Stalking Vengeance, Vicious Shadows, or Warstorm Surge is in play already.

If there's some other sacrifice outlet in play, such that the Crafter doesn't have to use its ability on itself, it could deal damage on every loop instead, creating infinite damage.

With Kiki-Jiki, Mirror Breaker out, and your opponent has no first-striking creatures, cast Combat Celebrant (Celebrant 1). Use Kiki's ability on the Celebrant 1 to make a copy that has haste (Celebrant 2). Move to attack phase, and declare Celebrant 2 attacking (it has haste, so it can), and exert it. That immediately (before even blockers are declared) untaps Kiki.

The opponent may declare blockers and may destroy Celebrant 2 (that's fine, as long as the Celebrant did some damage to something). In response to blockers being declared (attack #1) use Kiki's ability on Celebrant 1 again, creating Celebrant 3. After attack #1 is done, due to Celebrant 2's ability, there's another attack phase. Declare Celebrant 3 attacking and exert him. This immediately untaps Kiki.

This process can repeat infinitely; as long as your opponent has no creature with first-strike capable of destroying the Celebrant without it doing damge first, or has some indestructible creature (or protection from red creature) that can sponge up all damage, that should be enough to end the game right then. If your opponent does have something indestructible, having an additional creature with some sort of evasion (flying, likely) that can keep pinging the opponent would also end the game.

Note that when using the Sundial of the Infinite in combination with Kiki-Jiki, Mirror Breaker or Mimic Vat, the ideal time to use the Sundial's ability is right after the start of the "end turn" phase. At the beginning of that phase, it puts a "you must sacrifice this token" effect on the stack. In response to that taking place, activate the Sundial. The first thing that "ending the turn" does is remove from the game all things on the stack. The "you must sacrifice this" is not a spell, so nothing physically gets put in exile, but the triggered action does not resolve. Therefore you don't have to sacrifice the tokens, and the game continues. (ref)