are both connectors damaged? both the hdmi and the av connector? how did you manage to do that? make sure that there are pins short circuiting in either or both of the damaged connectors. this might cause an issue since you report video output problems. what does your xbox do when you try to boot without an cable attached? you should get a red ring indicating that no av cable is connected. if you don't get that, some pins might be shorting letting the xbox believe that there is a cable connected and outputting garbage due to short circuiting of the remaining pins.
you can always replace the complete damaged connector
or
solder the outputs you want directly onto the motherboard. you can solder a vga connector and RGB outputs directly onto the motherboard, these connectors use the same pins and are easy to solder. drill a few holes in your 360 casing, and glue those connectors onto the outer casing. make sure the wires are long enough, in case you have to take your mainboard out again in the future.
i myself have a garbage xbox, i am in the process of soldering the wires and connectors in directly (component, composite, vga, audio analag and digital). wiring diagrams can be found here on the forum, most probably in one of the stickies....