I have had my iPad mini for a long time, and it's been having issues for the past year. Whenever I let it completely loose power, it takes forever to actually turn back on. When I have it plugged in, it will power back on and show the apple logo on the white screen, and then shut back off. It does this a few times (like 5 or 6 times) before actually powering all the way on. Does anyone know what's going on or how to fix it?
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).