The Face, but with legs. He has 1/2 the stats The Face has, but much more speed. Leggsy is similar to Shy Boy due to his nature of attacking continuously and fast speed, but drops as much money as The Face upon death and has one more knockback than Shy Boy.
The best way to defeat Leggsy is to treat him as if it was a Shy Boy, but without any specific Red/Floating weaknesses. Since he is Traitless, Leggsy has far fewer crowd-controllers and debuffs that can slow his progress. Instead, use plenty of meatshields with general attackers and if that isn't enough, weaken him with Magica Cat.
When his running and chomping is put to an end, Leggsy will drop as much cash as an ordinary Face does, so don't be afraid to use a lot of your budget to kill it. Although, this is not that possible in Floor 18 of the Heavenly Tower, where you will probably be tight on cash until you kill the first or second Leggsy.