Apple Pencil not working even when connected and charging.

The double-tap on the Pencil body is does not rely upon the transducer - the transducer interacts with the iPad screen hardware.

If you have access to another iPad, compatible with your second generation Apple Pencil, you can test your Pencil with an alternative device. If the Pencil works, then you can look for issues with your iPad; conversely, if the Pencil does not work with a different iPad, you have confirmed that the Pencil is broken.

Apple Pencils are sealed. Other than the replaceable Pencil tip, a broken or malfunctioning Pencil cannot be repaired - only replaced.

You would be well advised to take your Pencil along to your local Apple Store - where your Pencil can be tested by the technicians:

Genius Bar Reservation and Apple Support Options - Apple

The Pencil tip was broken off?

Beneath the user-replaceable plastic Pencil tip is a metal transducer. This is a fragile component upon which the Pencil relies. If the fragile metal transducer is distorted, damaged or missing - the Pencil will require replacement. Repair of an Apple Pencil is not possible.

If your Pencil is within its one year warranty, it may be worthwhile asking for help at an Apple Store - but be aware that accidental damage is not covered by the limited warranty. However, if your iPad has AppleCare+ cover, then accidental damage of the Pencil may be covered.