Why do I need Wifi enabled for Kid Control?

Wi-Fi has two functions:

1) Location tracking: through local networks, without knowing their passwords, KidControl receives location coordinates.

Yes, KidControl is able to acquire location from password protected Wi-Fi networks, thus always keep Wi-Fi connection enabled.
After acquiring location coordinates from password protected Wi-Fi networks, the app needs to send this data to the server. For this, a working mobile Internet connection (data plan: 3G, GPRS, EDGE, UMTS, WCDMA, HSDPA, HSUPA, HSPA, HSPA+, LTE) is required.

Accuracy of location acquired via Wi-Fi is about 20 meters.

2) It provides Internet connection: through networks with known password, the app sends information about phone location and its battery level to the KidControl servers.

Mobile Internet (data plan) is not used then.