Apple Pencil 1st gen won’t start

The battery in the pencil is probably reached the end of its life. The batteries in devices these days do require a little bit of maintenance. If you had the pencil sitting around for a couple of months and didn't use it or try to use it and put it on the charger, then the battery was probably depleted to a point that it will no longer accept a charge.

When you now plug it in, even though you leave it on the charger for a while, the battery probably can no longer accept a charge and that's why it's not responding. Your only option at this point is to replace the pencil if you are looking at using one.

You can also bring it in to an Apple Genius Bar or other certified repair center and have them take a look at it to verify.

Be aware that if your Pencil has been set-aside for an extended period, unused and uncharged, the Pencil battery may have been damaged through deep discharge. In common with all Li-ion batteries, they must always have some level of charge.

The tiny internal Li-ion battery is 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.

Gentle warming of the Pencil (such as placing on a hot water radiator), prior to attempting charging of the Pencil, can sometimes resurrect a dead Pencil battery.