Contemplating whether or not iPhone would be worth repair?

I have an iPhone 4 Rev A (8GB) with a damaged LCD and digitizer. The phone has no warranty, so Apple wouldn't be able to help me in this situation.

When connected to iTunes, the iPhone responds well - everything shows up, such as my phone number, serial number, etc - I can even restore the device. However, strangely enough, the phone seems to be completely unresponsive. Despite the damaged LCD, I can still see the backlight which stays on indefinitely until the device's battery goes flat. The lock button doesn't seem to turn the screen off. Another problem is that when plugged into a power source, the charge noise seems to repeat constantly every 15 seconds, whether the iPhone is on silent or not. I can't take screenshots or anything either.

The phone is activated, freshly restored and, according to iTunes, is running fine. I can even install apps and music to my device. But the phone itself is acting very strangely. When I try to call the phone, it rings for a second and then instantly goes to voicemail. The phone doesn't vibrate when the ringer switch is toggled. But, the charge noise can be heard through earphones if they're connected.

I do believe there is life left in the phone - would it be worth getting the front panel replaced? Or is this phone dead? The phone worked perfectly fine before the LCD and digitizer became damaged, and for a little while after too. But after a week or so, when it'd been sat in my drawer for a while, this has happened. I've also checked the liquid indicators, and they're all still white, so I'm guessing the phone hasn't gotten wet at all.

Is my phone worth the repair?