IPhone 4 stuck on black screen after Jailbreaking with Cydia

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.

Plugging a phone into a computer or power source via the usb cable will power it on, even if it is off – your phone is in need of repair or replacement.

I've seen the apple logo appearing for a few seconds and then phone dying several times with dead phones. I've fixed countless phones, but I don't have a concrete answer for this problem other than it being related to the motherboard in your phone and the solution is a replacement (replacement part more than value of the phone)