As long as the device works with a computer you should be fine, But if connectors are broken there is a good chance it wont. iPhone needs to work normally, The only exception will be the screen.

Was it recorded with the phone or was it filmed and then put onto your phone?

An Iphone can still operate with non of the ribbon cables attached. I know this because one of the iPhone's I'm dealing with (3G) is fully functional without 1 2 3 connectors infront of me. Yesterday the connectors ripped out and it rendered the phone lifeless, so I left it on charge for a few hours and heard it vibrate back on. It must of malfunctioned or something. Now my computer detects it. A good test to see if it is phone is operating flick the mute switch on and off, if it vibrates there's a good chance your videos are recoverable.

Kerr McGuffie, first off, the iPhone does not have a hard drive. It uses hard wired memory IC's to store all the information. Bad part is, these are only accessible via special equipment. The silver lining on the horizon, if your computer still detects your iPhone as any kind of drive, your information is recoverable. For now, I would just make sure that you turn auto sync in iTunes off, and see what happens when your plug your iPhone in. It is most likely that it will work despite the broken cable. "The phone was not taking a charge" could give you a bit more trouble. Could be as simple as a bad battery, but check the phone pout first. Hope this helps, good luck.