Turning phone on if lock button is broken?
If an iPhone is off, and you connect it to a power source (either a charger or computer), it will always turn on. Since you have mentioned that this is not happening, I would conclude that there is something more serious wrong, not just that your power button is broken.

It may be the case that your phone has just 'locked-up' - try holding both the home button and power button for at least 30 seconds and see what happens.

If that fails, I would suggest taking it to an Apple Store and see what they say.