High cadence is harder on the lungs than the legs, so forget cadence (and numbers, strava, etc.) for now and focus on your weakness. If you want to go faster you'll need stronger legs, and quickest way to do that is practice going faster. You can increase your leg strength a lot quicker than you can attain a perfect pedal stroke. Just go all out for 2-3 days a week and give yourself time to recover with easy alternate days. For a 7 mile commute, it should be easy to get to your 18 mph goal within a month or two.