My Ipad Mini 3 doesn't turn on

Start by connecting your device to an iTunes enabled computer; does it get recognized? If it does, then you have some kind of screen related issue.

If it doesn’t, then the device is not booting properly, which given the age of the device, is probably battery (or charge system) related. When troubleshooting charging issues, I would do the following (in order):

1. Always try another Lightning cable and charger, preferably Apple original or MFi certified units.
2. The Lightning cable 8-pin connector should be sitting flush with the housing and completely inserted. If it is not, there could be lint/dust/debris inside the port impeding a proper connection. You can clean it out with a toothpick, fine point tweezers or a dental pick. Just be careful not to damage the pins inside the Lightning Port.
3. Change the battery. The battery is the weakest link in the entire device and certainly for charger related issues - especially on older devices that still have the original battery. Please let us know what voltage you measured at the battery. The iPad requires pretty high voltage to boot properly.
4. If the battery is in good condition, you can use a USB Ammeter to determine if the device is really drawing current when it connected to a charger. If it isn't drawing current, then I would change the Lightning/Charge Port.
5. If a new battery and charge port still don't resolve the issue, then you either have faulty replacement parts (possible) or a logic board issue. There is an IC (commonly referred to as Tristar) responsible for communicating with the charging device and uncertified chargers can damage it. Unplug the battery and connect a known-good (preferably Apple original or MFi) charger to the device. A properly working device will show an Apple logo and boot loop. If the Tristar IC is bad then the device will not do anything.