Before replacing the entire ignition switch, disconnect the scooter’s batteries and try to clean it. You can use a pressurized can of air to blow dirt and grime out of the ignition, which may be enough to fix it. If that doesn’t work, you can order a replacement ignition and install it yourself; it takes a lot of time to do, but it isn’t an overly difficult repair if you have the patience!
Removing a broken key is easy with a pair of pliers. However, you may want to leave the broken piece in place until you get a replacement key. That way, you can use the pliers to turn the key and use your scooter while you wait for the replacement in the mail!