That's a tough situation to be in. In my experience its unlikely the magsafe board. Most of the time its ends up being an issue with the board. This may be as simple has cleaning corrosion on the board that is bridging and shorting some circuitry or it may be more involved where you would have to find the plans for the board, troubleshoot connections by checking continuity in search of failed components. Not to mention that the job becomes a bigger hassle without the proper tools. If you have to the time, patience and bravery, there are a few resources online that can help you get there. iPad Rehab is a company that teaches people how to get started as well as offers repair options.
I would start by trying to isolate hardware from software issues. Best way to do that is to boot the machine to an freshly installed OS on a bootable thumb drive. If the issue persists its a hardware issue, if not you have isolated it to a software issue.
Matt, if you can see the full image on the screen, then the only damage the ipad sustained was the glass. If there are any visual imperfections in the image quality such as lines, discoloring, color bleeding, and dark spots then the LCD was damage as well.