4x4 light on and the digital display is off | Dying battery wont hold charge

2013 S 4x AT | 35A Battery - Trickle Charger does nothing, battery is toast.

I'm in between jobs and trying to scratch together enough money to get a new battery at the moment. I have been jump starting the car with my battery pack jump-starter but the 4x4 light comes on and the digital display that shows which mode the car is in is blank. Drove the car about a quarter mile and the 4X light did not turn off and the display never turned on. Once the car is running the battery indicator jumps up to normal.

I have a job interview next week on the other side of town, about 20min away via the highway and I'm guessing it would be a pretty bad idea to try and drive it that far.

Thoughts and feedback needed. :/