iPad broken digitizer connecter, want to be able to control device without screen temporarily

What the title says. The last person to fix my broken screen really wasn't taking enough care of this thing, so screw mr. repair guy. He left out 5 total screws and somehow loosened the digitizer connector on the logic board itself. So when I try to fix it myself, it comes clean off. There's the situation.

As it stands, I am fully unable to use my iPad for 2 reasons. First, the fact that my computer cannot "trust" it without me signing into it prevents me from connecting it to iTunes, and second, I can't install any apps without the touchscreen.

The reason for me wanting to be able to control the device is because I haven't backed it up yet, and I have a bunch of important stuff on it that I would like to transfer to a new device once it gets here.

Answers..?