First of all, do not sell yourself short! Jogging can definitely be considered running. Like you mentioned, it might not be the same pace as running but it certainly requires an incredible amount of self discipline, training, and mental focus to jog any amount of distance. Furthermore, do not fall victim to thinking that jogging does not have the same benefits as running. From a health stand point, any amount of sustained cardio that increases your heart rate will result in a healthier heart, more calories burned, and improved long term health effects. Plus, a slower paced jog that does not require as much physical (or mental) effort might be just what you need if you have a stressful day or are just generally anxious and need some relief and endorphins.
To increase your speed, you should first focus on increasing your mileage. Building up your endurance and stamina will mean a base for you to increase speed from. Slowly increase the number of miles you run each week by adding on an extra 1-3 miles each week. You can also increase the number of days you are running, with the goal to run every day for five to six days a week (being sure to take breaks when needed and always at least once a week to give your body time to heal and recover).
Next, incorporate speed drills into your workouts. These usually fit best before or after a workout, and include drills like high knees, butt kicks, and carioca. You also want to be sure to stretch well, because speed requires limber and well stretched legs that can stride and lengthen out. Strength training is also a huge component of increasing speed because you build up the muscles in your legs that propel you forward faster, and the muscles in your arms that help you pump them faster - all of which work together to achieve a faster leg turnover time and thus, faster paces.
If you take away anything from this to increase speed it is to always concentrate on nailing excellent form, and willing your body and legs to work as hard and efficiently as possible.