iPad mini won't turn on/charge
Unless I'm missing something, this is expected behaviour when your iPad battery is extremely low. Basically, the battery will need a certain amount of charge to allow your iPad to switch on, and a certain amount of charge to effectively boot up. 

If your iPad mini works fine otherwise, then it's really probably just a case of not letting it get too low before placing it on charge.

Also, you need to ensure that you're connecting it to its original charger, or to a charger providing enough current to charge the iPad effectively. In the absence of doing this you may find the iPad takes a lot longer to charge before being able to boot it up again.

Finally, as you've had your iPad mini for a long time, the battery may have exceeded its expected charge cycle count. When this happens it won't maintain a charge for as long as you're used to and it will take a lot longer before it can reach a state of charge needed to work properly. Eventually, it will die altogether (although if it has enough charge then you can effectively use it while it's connected to a power source - but if it loses that state of charge, then even having it connected to a power source may not suffice).