If there truly is no sign of life at all -- no sounds of any kind, for instance -- then a fried power supply is likely. In your case, with a badly corroded battery, there is a chance that the on/off switch has suffered corrosion as well. With the unit unplugged, try rocking the switch back and forth some number of times -- that might clean off the contacts enough to get something to happen.

In any case, opening it up and having a look around will be highly revealing, and almost certainly necessary.