rom – Need to flash Galaxy Tab PRO 10.1 to recover data

I hope you can help with this problem:
Samsung Galaxy Tab PRO 10.1 SM-T520 Kitkat 4.4.2

This is my wifes tablet and it recently stopped booting at the second Samsung logo.
It's the one that comes after the purple swirly (sorry for the technical term).

I went through a few simple things to try to fix it, including reboots.
I've also been disconnecting the battery for a while while pressing the power switch.
I hoped that it could solve the problem, but it was not the case.

I tried to connect to my wife's computer where he was not recognized (I was working there before).
I've also tried my system, I did not have any software installed, but I tried the latest Kies software.
I've also tried an older version of Kies and also the Samsung Smartswitch. Later, I also tried Odin.
Nothing recognized that the tablet is connected. Tried several cables in vain.
If I look in the hardware manager on both computers, no tablet will appear.

Looks like the USB tablet is dead or the software does not work far enough to activate it.

Has researched and entered the recovery mode, here is what presents itself.

Android system recover <3e>

Dead android with "no order" below

In my research, I discovered that you can update an image of the SD card.
So, I found two US roms (XAR region) for Kitkat 4.4.2, I tried both.
I used "apply the update from an external storage" in the recovery menu and selected the ROM.
They seem to start loading / extracting on the tablet and then I get the following;

E: the footer is wrong
E: failure of signature verification

The above message occurs twice. The only reason I did not just use the factory settings is because my wife has not backed up the tablet in the last 6 months and the previous backup contains empty directories. Although nothing is critical on the tablet, she saved a lot of quilting patterns / free clothes.

So, whatever I do, I try to preserve the data, hence my efforts with the ROMs.

I can not change any settings on the tablet, I can not run the USB stick and I continue to have errors with ROMs. I have already seen it suggest that slow downloads can corrupt the zip files I download. I do not want to pay for the download. Is there a way to get free ROMs at a decent download speed to eliminate this problem?

Everyone has ideas for everything I can try.

Thank you