You'll increase speed naturally as your (running) fitness improves. Pushing yourself to exhaustion every day will lead to breakdown rather than improvement. As running is a high impact activity, there is far more stress/muscle damage to recover from.
Two hard workouts plus a longer run per week with non-impact activities on the other days will be more optimal for good progress.
Your idea of increasing pace throughout the run is fine, it's the way I used to train when I ran.
"To give anything less than your best is to sacrifice the Gift."