I am not a swimmer but after searching on many website vertical flutter kicking is the best technique for improving kick and make kicks begin at the hip.
Article on triathlete has a conversation that mention these two drills for focusing on strengthening your core, glute and hip extensors.
Extend Leg Drill
Swim with one leg extended out of water, then switch to the opposite leg after a 25.
Vertical Kicking With Weight
Try vertical kicking sets holding a 1–5-pound object out of the water (10 rounds, 45 seconds kicking, 15 seconds off). “During this set, the legs are forced to kick with equal force and direction, while the core and shoulders contract in unison,”.
Article from azcentral
Competitive swimmers typically utilize the two-beat cycle, kicking two times for each arm cycle -- essentially one kick per arm stroke. Another option is the six-beat cycle, in which you kick six times for each arm cycle. This is also called the flutter kick. Because it's so fast, it can be tougher to incorporate the hips with the six-beat cycle.
and many websites articles mention vertical kicking as the best way to improve hip kicking. Links to other website articles UWhealth, US MASTERS SWIMMING, Swymnut Masters, SEAHIKER and Enjoy Swimming.
At last Article from SwimSmooth:
A good way to think about kicking from your hips is to squeeze your bum when you kick. It's a bit crude, but imagine you have a large coin between your butt cheeks and you're trying to hold it there whilst you swim. Keep your knees relaxed and drive the kick from the hips.
Note: All the quotes are copied from the respective website and i'm not original author. If I have failed to mention any copyrighted material that is clearly my mistake.