You have to get Kara's Obedience Stats up high enough where she'll do it.How to get
Kara to go to the Temple for the key?
The stats/harem system is still very much a work in progress. One of the things I'm hoping to implement in a future release is some kind of feedback on the stats and what you need when a stats-check fails.
Also, more ways to increase stats, such as Harem girl trainers dotted across the landscape.
In the current release, you can drive up stats like Kara's obedience by playing through the game world (many achievements just by completeing quests also increase stats) and finding and collecting certain treasures, which also will increase very stats. Some are items that increase stats across the board, others are one off gift that buff a single harem girl stat.