You can currently purchase a replacement earphone socket on eBay for $4.99. Search for something like "ipod nano 2nd generation ribbon cable". The earphone socket is attached to the replacement click wheel cable. Instructions for replacement are here: How do I replace the wheel assembly?
I had a white screen problem also. A close inspection with a magnifying glass revealed that I tore the rightmost edge (the image is upside down) of the display ribbon cable, and this is why the display turned white. The ribbon cable is fragile, tiny, really impossible to repair by soldering. I thought about using this Nano without a screen (somewhat like my iPod Shuffle) but ended up putting in a different screen that I happened to have on hand. It seems to be fully functional!
Before committing to the expense of a new screen, I would disassemble the stepped-upon part (presumably the screen) to inspect for damage. You might learn something useful. Maybe it is just the cable. Maybe some additional parts are also damaged and need to be replaced.
Your battery is not the problem. I tested my Toshiba Tecra M3 laptop and with the power adapter plugged in it will boot up with a flat battery or even with no battery at all. Because your laptop showed signs of life only with the new power brick, the old one is probably bad. Rather than throwing out the old power brick, you should troubleshoot it and see if you can repair it. When you power on your laptop, listen closely. You should hear the hard drive spin up almost immediately. If your laptop has been without power for a long time, it will beep and prompt you to press F1 to reset the date and time. If your laptop is completely unresponsive, disassemble and check the power button. While it is disassembled, you can inspect for signs of the spill you suspect that your kids committed. You can find a link to the Tecra M3 manual here: https://support.toshiba.com/support/mode... On page 209 it says this: The computer will not start. Make sure you attached the AC adapter and power cord/cable properly or...
Troubleshoot this by typing with an external USB keyboard to see if the problem goes away. If it still happens, then try reloading Windows. If the problem goes away with an external keyboard, then consider operator error. Undesired cursor movement could be caused by inadvertently touching the trackpad while you are typing, with your fingers or the heels of your hands. If you are sure operator error isn't the problem, then a bad connection in the trackpad could be causing the problem. Troubleshoot this by opening the laptop and disconnecting the trackpad ribbon cable. Test to see if the problem goes away. If I recall correctly, the trackpad ribbon cable can be tricky to reattach if you go from the bottom (I had this problem with one of my Toshiba laptops). If this is the case, do some obvious disassembly and you will easily be able to reattach it by working from the top.