So my lock or "hold" button is broken and my phone turned off. My phone is plugged in to the charger and still won't turn on. Also I can't plug it into my computer because my computer doesn't recognize an iPhone as a device.
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.
You probably need to some repairs to your hardware (iphone) Unfortunately I do not think it is you button (the plastic piece) but what is behind it.
Here are instructions where to get it and how to fix that
Is it easy to do - NO !