If your NDS can no longer recognise the game card you may have a faulty Slot1 card reader on your DS. These can be replaced.
I assume you already know how to take the Nintendo DS Lite apart.
A Youtube guide can be found here.
You will need a solder sucker or solder wick, I find that the solder wick works much better.
The revised technique doesnt require the solder sucker to remove the slot1, Solder wick is just required to clean the excess solder of the board.
Apply a bit of solder to the wick before you try to remove the solder of the solder point. This will make the solder thats currently on the board move to the wick much easier.
Once you desolder and sucked all the solder mounting points off, you should be able to lever each corner up at a time. Work your way from the top to the bottom corners. Once you get to the bottom where all the joints are attached to the board. You should be able to heat each leg and lift with the soldering iron.
There also the other method, you can the legs and then clean it with the soldering Iron. However there's a high chance of cutting the circuit and causing the board to be useless. I accidently did this - NOT RECOMMENDED.
Latest and easiest way I found to take of the Slot1 with out having to use the solder sucker or which is;
Start with the small legs ( abcdef...), working from left to right and using both hands. Left hand with a tweezer or something small and thin - hold this behind the pins.
The soldering iron in your right hand, heat the surface area of the first pin and use your left hand to lift the leg. Apply enough heat, do not force the legs - if not done correctly you will stuff the board up by pulling off parts of the PCB.
Work your way across untill all pins are done - abcdefg...... Then you do the four points, starting with 1 - 2 -3 - 4 order.
This is what it looks like once it is off. Re work the solder points on the board preparing it for the new Slot1 by adding just enough new solder. Do not over do it.
Tin the new Slot1 legs so that it will bond to the board better.
Tip : When testing, use an original cartridge - I have used a TT card and it worked and the original didnt. Found out that some of the pins were not soldered properly. Once rectified both cards worked.