Ipad Mini 3 stuck in recovery mode during update to 8.4.1

You can check this support document for an explanation of errors. Get help with iOS update and restore errors - Apple Support

I don't see error 18, but error 53 can be either a security software issue or a hardware error. Error 3004 has to do with software, but you can click on a link in the first document I provided to get tot the document that explains 3004.

If after following the instructions in the two support documents, if you are unable to update and restore the iPad, then make an appointment at the Genius Bar of the nearest Apple Store to have the device examined.

Was your Home button already "bad" before you updated? I know you mentioned that your Home button does not work but this is worth a try at least...note that you may need to try multiple times to get the device to be re-recognized at it may have locked up. I had the same thing happen a few times during iOS updates on different iPads and the forced restart (again, a few times was needed) always got them out of the lockup. But in my case, this was with a working Home button. If your Home button didn't work before the update, chances are good that you won't be able to force restart since that requires a working Home button. Give it a try anyways & good luck. If you can't get the device to respond, take the advice of ChrisJ4203 and take it in to the Genius Bar if you have access to one.

You'll need your Apple ID to make the reservation:

http://www.apple.com/retail/geniusbar/

Force Restart (won't affect settings/data/music/apps/etc)

1. Press and hold (& continue to hold) BOTH the Sleep/Wake button & the Home button.

2. Continue to hold BOTH (ignoring any other messages that may show) until you see the Apple logo on the screen.

3. Release BOTH buttons when you see the Apple logo and allow the device to boot normally.

The screen you are showing there is the screen to identify Recovery Mode. You can try information in this support document If you can't update or restore your iPhone, iPad, or iPod touch - Apple Support regarding recovery mode. If that does not work, your only option is to make an appointment at the Genius Bar. I don't know why you don't want to go to the Apple Store, this could be the only way to get your device corrected.

Well , if your Home button is indeed dead, then a Force Restart cannot be performed so you're stuck there.

Like Chris mentioned, you are in Recovery mode. You have already lost all content on the device. There no getting around that fact. If you had a backup, you could possibly restore from that backup after recovery. You mentioned you "tried recovery software to restore my iPad and nothing". Was this some third party software package or something or iTunes? Did you use iTunes itself to restore the device? As I mentioned, you've already lost everything on the device so you have nothing to lose by recovery via iTunes.

If you get the errors during recovery, try again, and again, and again. If no luck and you get the same errors in the same place, then you probably have no choice but to take it to Apple or have a useless piece of metal, plastic and silicon.