Macbook late 2010 13" won't charge?! Please help!

Question

Ok so my mom gave me her old Macbook (late 2010) 13" laptop after it stopped working for her. She claims she didn't drop it but there is a large hunk of plastic missing around the magsafe port and it will no longer charge and the DvD drive doesn't work either (only makes a very annoying sound upon boot or waking). I was looking into the usual charge cycles for this particular battery a while ago and I believe it was around 300 which I was about 50 over, so I went and bought a brand new Apple brand battery and installed it only to find that it, like the other battery, still won't charge. My Macbook recognizes it is there but just won't charge. It says that this battery only has one charge cycle which indicates that it is indeed brand new. The battery icon on my top bar used to say service battery until I replaced it, now it only says "Not Charging". Here are my battery specs and if you need any info to help me, PLEASE ASK. Someone please help me save this wonderful computer....
Battery Information:
  Model Information:
  Serial Number:          W00078AH89X7A
  Manufacturer:          SMP
  Device Name:          bq20z451
  Pack Lot Code:          0
  PCB Lot Code:          0
  Firmware Version:          103
  Hardware Revision:          000a
  Cell Revision:          130
  Charge Information:
  Charge Remaining (mAh):          4109
  Fully Charged:          No
  Charging:          No
  Full Charge Capacity (mAh):          5552
  Health Information:
  Cycle Count:          1
  Condition:          Normal
  Battery Installed:          Yes
  Amperage (mA):          0
  Voltage (mV):          11704
System Power Settings:
  AC Power:
  System Sleep Timer (Minutes):          10
  Disk Sleep Timer (Minutes):          10
  Display Sleep Timer (Minutes):          5
  Wake on AC Change:          No
  Wake on Clamshell Open:          Yes
  Wake on LAN:          Yes
  Current Power Source:          Yes
  Display Sleep Uses Dim:          Yes
  PrioritizeNetworkReachabilityOverSleep:          0
  Battery Power:
  System Sleep Timer (Minutes):          15
  Disk Sleep Timer (Minutes):          10
  Display Sleep Timer (Minutes):          5
  Wake on AC Change:          No
  Wake on Clamshell Open:          Yes
  Display Sleep Uses Dim:          Yes
  Reduce Brightness:          Yes
Hardware Configuration:
  UPS Installed:          No
AC Charger Information:
  Connected:          Yes
  ID:          0x0000
  Revision:          0x0000
  Family:          0x0000
  Serial Number:          0x00000000
  Charging:          No

Answer 1

A brand new battery would require a full charge in order to be used;
so you are lucky, in that the new battery had enough power already.
So that is not an indicator of much, other than the MagSafe Board
in the computer itself may have sustained damages due to ???
and would need to be checked over by an authorized tech or if you
are near an Apple Store with Genius bar, call ahead for appointment
to have a diagnostic test done and take along the MagSafe adapter
so they can see if that is part of the problem.
Of course, an optical drive issue could be part of a diagnostic check.
If you are in the US or Canada, the retail Apple Store offers a free
service of a diagnostic, even though the computer is no longer
covered by an AppleCare plan. So that is worth taking a drive...
Check the details here, and follow them to see if you can make it go:
•Intel-based Macs: Resetting the System Management Controller
http://support.apple.com/kb/ht3964
•About NVRAM and PRAM:
http://support.apple.com/kb/HT1379
Are the contacts on both the MagSafe adapter and MagSafe port
clean, or do they have an odd color or debris in/on them? It is
possible to clean them of debris, but don't use watery stuff. A
tool such as a wooden toothpick could be used to see if anything
on there could be rubbed off or out of the contact areas.
Good luck & happy computing!