There are two ways to go faster:

1) Pedal faster (higher cadence, usually lower gears),
2) Mash harder (lower cadence, higher gears).

1) Uses your heart and lungs to increase speed,
2) Uses your legs to increase speed.

To go faster, you need to improve the fitness of the systems you're using. If you're a spinner, you need to work on your heart and lungs. Spinning is a good way to do that, but any cardio activity will help, including treadmills, walking, running, and elliptical workouts and you will, eventually, begin to realize performance increases. If you're a masher, you will build leg strength, and any exercises that assist with leg strength (squats, leg extension machines, stair climbing) will increase the strength of your quads and glutes, giving you more muscle mass to push those pedals down.

Ideally, you'll work towards a balanced approach, using high cadence spinning where appropriate (long, tortuous climbs), and mashing when appropriate (just got passed by a girl!).

In the end, if your heart and lungs can't supply fuel and oxygen, it won't matter how strong your legs are; you'll still fade. It may not seem like much in the way of help, but you need to work on everything, including your upper body and core, to be the best cyclist you can be. Do the exercises that help you to lose weight, but mix in some strength and cardio, too.

I think your goal of 27 kmh by the end of the year is reasonable, given regular work, a decent mount, and no headwind! Go for it!