Using a damaged or otherwise defective Power Adapter or charging cable is unwise - as this can cause damage to the iPad. Good quality (and ideally Apple) Power Adapters and charging cables are relatively cheap when compared with the high cost of repairing an iPad.
As described, your iPad battery may have a fault, or may be reaching the end of its usable life. If the battery has failed or is failing, the iPad may unexpectedly turn-off - or may not successfully boot to a running state even when connected to its Power Adapter.
Are you able to use the iPad while remaining connected to your Power Adapter? If yes, while monitoring the indicated battery state, disconnect the Power Adapter; if the indicated battery state suddenly falls, or the iPad shuts-off, this test demonstrates the iPad battery having reached the end of its usable life:
iPad Repair & Service - Apple Support
You might be well 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, Power Adapter and cables:
Genius Bar Reservation and Apple Support Options - Apple