If the adapter cannot be recognized, and make sure it is a genuine Dell OEM charger,  it is usually 1 of 3 problems. Either the adapter is defective, the DC jack is the cause with a broken or bent centre pin, or the MB itself has a problem. From your post is does not appear to be a battery problem but rather everything points to the adapter.
Dell inspirion 15 5565 Does not recognize battery and charger.