Your Apple Pencil is exhibiting symptoms of a failing/failed battery - that will not reliably accept or hold any charge.
It is generally recommended that a second generation Apple Pencil should be kept docked with the iPad when not being actively used - as this ensures that the tiny internal battery is kept optimally charged when the Pencil is not in use. Charging only occurs when the Pencil battery requires charge from the iPad, optimal charge being entirely automatic.
Keeping the Pencil separated from the iPad when unused risks allowing its battery to discharge; if allowed to remain in this state, the battery will suffer irreparable damage. When the battery fails, the only remedy is to replace the entire Pencil; the battery is not a serviceable item.
Li-ion batteries are susceptible to permanent/irreversible damage through being left discharged for long periods. Even some “new” pencils can exhibit signs of failure out-of-the-box if they are “old stock”. It is essential that if you have an Apple Pencil that you charge it regularly - whether used or not - so as to protect the battery from deep-discharge. Do not allow a pencil to remain in low-charge state for any period of time - as the internal battery will fail, rendering the Pencil useless. Setting aside an unused Pencil, for extended periods, is a recipe for premature death of the Pencil battery.
If the Pencil Battery has failed, the only remedy is to replace the Pencil. If the battery has failed and your pencil is within its one-year warranty, you should look to having it replaced by your retailer or at an Apple Store.
You would be best advised to visit the Genius Bar at your local Apple Store or Apple Authorised Service Provider (AASP) - where the technicians will be able to test and assess your iPad and Pencil. If a fault is identified with either, you’ll be in the best possible place to explore your available options:
Genius Bar Reservation and Apple Support Options - Apple
