I have no DSi to pull apart so I am unable to call it but that does help a bit
Forward Voltage (V) : 3.2 ~ 3.8 (quick aside- that is a rather low maximum voltage but a decent operating voltage and your quote does not have another value for maximum so until I know otherwise I am going to assume 3.8 is the maximum)
This means it should glow green at around 3.2 V- possibly a bit higher than the DSi LEDs you are replacing but 3.3V is a common voltage to work at inside circuits.
Re resistors: yes you will need for "raw" batteries/power sources but the designers of the DSi also know this so there is likely a resistor already in the circuit. You may however find this pre existing resistor to be wrong for your needs so you may have to remove or "bypass" (with surface mount tech this usually means solder over it/bridge it) it.
As it pertains to your LEDs there are three main things you want to know (in addition to the operating voltage above)
Max Power Dissipation : 80mw
Max Continuous Forward Current : 30mA
Max Peak Forward Current : 75mA
Everything else is useful for the would be electronics engineer but for your purposes can be ignored.
75mA is the absolute maximum- any more and it will probably burn out. You need not concern yourself with the really involved stuff like capacitor discharge pushing over that limit (what that rating is most likely to be used in) as you can assume the DSi designers took care of that.
30mA is the maximum for the "ideal" current. Assume it is running continuously (power, charging, wifi and the like) so do not be tempted to "break" the spec.
Power Dissipation you can ignore for the most part unless it turns out you are running at a slightly higher voltage at or around the Max Continuous Forward Current- 80mW is just below it so just make sure it is not free floating in air or it will overheat and destroy itself sooner rather than later (or just stick a resistor in there). A quick aside is that those seem to be your standard bulb LEDs while the DSi would most likely use a surface mount one, while it will not trouble the electronics side of things so much it might not look all that brilliant.
http://www.ledshoppe.com/Photo/Product%20photo/PICT0155.JPG is what appears to be coming in your chosen LEDs while the DSi ones will look something like
http://media.digikey.com/photos/Lite%20On%...TST-C150YKT.jpg
I dare say you also want to read something like
http://www.kpsec.freeuk.com/components/led.htm as it covers the general theory behind LEDs and how they play in circuits.
You replace the LEDs where you found them unless where you found them does not meet the requirements (when replacing fans with higher end ones I will often draw from a point that is at a higher voltage or a point able to dish out higher current- it is part of the reason I mentioned that it might be higher). Naturally for something like this if you have to source power from another point then you will need to stick a transistor in there to get it to switch on and off as necessary.
Re: Wii-motes the logic/practices used here follow for all electronics.
I know I have just thrown lots of electronics at you but for the most part this sort of thing is as brutally simple as you probably thought it would be as you set out to do it- you locate the LED, you check to find the polarity (LEDs have a positive and a negative side remember) and you swap them over observing said polarity. The extras I mentioned are still important and should not be skipped over but take it a step at a time and you will get it done.
