My HP mini 2140 will not energy up, even when a AC adapter is attached. It has been sitting over a month, as well as I’m guessing which a battery is discharged. The battery light shows "amber", indicating it is recharging. Can a battery be so liberated which it will not energy up even when trustworthy to a AC?

No, if it is attached to the AC it should work fine.
Run these steps:
Unplug the AC.
Remove the battery.
Hold down the power button for 60 seconds (with the ac and battery removed).
Plug in the AC. (Do not put the battery back in).
Press the power button.
It should boot up. When it does, put the battery back in.
EASY FIX
STEP 1: GO OUTSIDE GET IN YOUR CAR.
STEP 2: DRIVE TO BEST BUY.
STEP 3: PURCHASE NEW LAPTOP, BECAUSE YOUR OLD ONE JUST DIED.
Replacement ac adapter 18.5V 3.5A 7.4×5.0mmB for HP mini 2140 at $15.99
http://www.chargerbuy.com/replacement-hp-65w-smart-ac-adapter-185v-35a.html
Hi 182Pilot,
You would need a new charger. Even if the battery was drained or dead, it would work with the charger attached since it uses power directly from that. You can purchase yours from Ebay or Amazon (most common) but I prefer TopMic. Their service is wonderful and I highly recommend them to anyone. I’ve been using their chargers for over a year now and have had no problems.
http://www.topmic.com/hp-mini-2140-charger.html
I took the liberty of going to the site and searching yours up. There it is.
Good luck! I hope you found my answer helpful!