Run an ePSA diagnostic and check it out.

The bios should say the battery is working perfectly.

Make sure you reinstall the battery drivers in the device manager.

Why wouldn't it beep indicating power rail failure though?

If the system does recognize and charge the battery but won't run from it, chances are there's damage to one or another of the power rails on the system board.  Unfortunately, this will either mean finding a shop that can diagnose and replace the faulty parts (which many shops are reluctant to do in the absence of schematics), or replacing the mainboard.

Depending on the revision of your system and the CPU and GPU you have, the board will run between about $300 and over  $600.  A repair with diagnosis through a skilled shop will likely come in between those numbers, depending on what's needed and whether they can find parts.

So Power rail failure? Then why wouldn't the laptop start beeping indicating a power rail failure.

Hi @Tu Doan a silly question, is the new battery an official Dell replacement? Look at the battery's label to check that it says Dell. If yes, Dell has a battery troubleshooting guide Here (includes check that Battery is recognised by the PC). 

The board has to be able to power up far enough before you'll hear a beep.  If it's a main CPU power rail, that may not happen - and CPUs from the last 10-15 years have multiple power rails, with varying voltages.  Every modern CPU does.

But since it has multiple power rails why would one make the entire thing refuse to charge.

The original post does not reference a difficulty charging.   In fact it states that the battery is 100% charged.  It specifically says the system won't run on battery power, which is why the power rails are suspect.

Teach me your knowledge @ejn63.

You need to teach me your knowledge @ejn63.

I'm studying my comptia and need to know everything about power rails and other hardware related components.