Symptoms
According to the following KB article regarding the K2 Mobile App, badges were implemented during the July 2014 release of the app: http://help.k2.com/kb001591
Configuration in the Notification center is not showing up for the K2 Mobile App in order to enable the app badges.
Diagnoses
This seems like a bug, as it was reproducible on several different iOS devices with the latest OS update (8.1.3). However, other devices allowed for the configuration of the K2 Mobile app notifications and badges without a hitch.
Resolution
The following bug was logged for the behavior:
Bug 524124: iOS - Badges not showing
This item should be fixed in the upcoming release of K2 Mobile. Unfortunately, there is no way to easily fix the current behavior.