There are many ways to get there, as your investigations would suggest. And perhaps you will be pleased to learn that long-endurance work is not the most effective way to achieve high middle-distance performance. If you do not enjoy longer runs, there is no need to do them.
Running speed is a function of stride length and stride rate.
speed = stride length × stride rate
This implicates two fundamental areas which may need to be addressed: hip mobility and leg turn-over rate.
Limitations in hip mobility are very common amongst amateur runners—I would argue standard. This is particularly true of hip extension, which is limited by the hip flexors. The femur of most amateur runners is limited to between 0 and –5° (behind the line of the torso), with the pelvis and lumbar spine usually compensating to permit greater functional mobility. The latter, of course, can lead to lumbar soreness and injury.
In the photograph below, we can clearly see the ease by which the knee of current 5,000- and 10,000-metre World- and Olympic-record holder Kenenisa Bekele (pictured right) passes behind the line of his torso to an angle of perhaps –30°. His torso is erect and straight with a slight forward lean, and there is no lumbar hyper-extension whatsoever.
Good posture is critical for speed, economy of motion, and injury prevention. If your running posture is sub-optimal, you would benefit from dedicated stretching of both the hip extensors (gluteals and hamstrings) and hip flexors (particularly the iliopsoas and quadriceps femoris groups). Stretches should be dynamic prior to running, and static afterward.
The second part of the equation—the rate of leg turn-over—is addressed through the development of speed skills. Ideally, most of your training should be performed above or at target speed (that is, 4:00.0/kilometre pace), and the distances that you run should reflect that. Effort should be made to lift the knees, to strike the ground beneath (not in front of) your centre of mass, and to keep your stride full and even.
Since you are already able to reach and maintain your target speed, you should limit yourself to distances that allow you to practise at (or above) that speed. I would recommend beginning with 2- or 2.5-kilometre bouts, and to aim not to drop below 3:50.0/kilometre pace, for example. As the pace becomes more comfortable, and as you achieve that pace more consistently, simply extend the distance in small graduations until you are running the full 5 kilometres.
I hope that helps. Good luck!