If the camcorder has been sitting under heavy stuff, or handled violently. The power switch tends to go inside of it's socket due to the access force being applied and the age o the plastic holding it in place, that causes the power switch to be unresponsive . To solve that issue:

Take a pair of tweezers or small plyers and wrap around the prongs cotton pads or paper to avoid scratching the camcorder

With the now cushioned tweezers / plyers grab the switch from inside of it's socket and gently pull it out until you hear a click, the switch should be back into place.

After that repair, be gentle with the switch because it's hinges have already been worn out quite a bit and unfortunately there is no way of fixing them.