It's is not dangerous.
First I'd learn to solder if you don't know how to yet, watch alot of videos and buy a good soldering iron with solder.
Go to dealextreme where you are buying that socket from and buy this:
http://tinyurl.com/3eh7c5g which will enable you to unscrew the special screws that nintendo uses.
Turn power to your system off.
Unscrew all screws. Having trouble? Here use this guide:
http://tinyurl.com/43weq8j
Once all the screws are undone take the battery door off (located to the right to the system in the above guide) and take the battery out. It is also a good idea to discharge yourself somehow before doing so.
No take of the back shell of the NDSL and you will be faced with a motherboard.
Locate the broken socket and heat up all the solder joints and get rid of the solder from them so you can remove it, this tool can help you get the solder off the board:
http://tinyurl.com/439bywe
Now solder the new socket to the board.
Make sure your solder joints are okay and put the back shell back on.
insert the battery and the battery cover.
Before adding all the screws to the holes hold the system together tightly so the battery doesn't fall out and test if it will charge with a charger:
http://tinyurl.com/3g8ryor
Note that if you power on the system and take the battery out when it's on that it will short circuit and won't ever work again.
If it doesn't charge then see if your solder is touching any other components and if it is desolder it and solder it back on so that it is touching other components.
Also if you need more help please ask.