You can check a lot of the internal wiring with an ohmeter.
First remove the battery pack and switch the unit on. Next measure across the outer two posts in the charger port. They should show zero ohms becuase they are connected together by a strap when the charger is unplugged... one pin is connected to the battery and the other connects to internal circuitry. If the strap is not making contact the machine will not run on a battery, but should run on a known good charger.
If the machine runs on the charger and the strap is making good contact but will not run on a known good battery then you probably have (often invisible) corrosion on the battery contacts. The battery pack makes contact to the terminals along the SIDES of the tips of the gold contacts. Clean well with very fine sandpaper.
Next measure between the center charger port pin and the negative battery contact. This should always show a connection.
Next measure between one of the outer charger port pins and the positive battery contact. This should show a connection. If not then the power switch probably needs cleaning.
Classic series chargers are notorious for bad connections in their plug. Also well known for a bad main electrolytic filter capacitor. Either problem can cause the machine to not light up.
If you need to open the machine to clean the power switch you can do it without removing the label. You need to peel up the top two corners of the label about 1/2 inch to access the screws. I wet a razor blade with the temporary adhesive releaser called UnDu and slide it under the corners. Be careful not to slice the label or sharply fold it up. Often the label corners will stick back down. Sometimes you need to glue them to keep them down. I never reinstall the screws under the label so that future repairs do not risk further damage to the label.