Turns on screen comes on and goes straight off

This is exactly what happens when either; A: your Ds lite's battery is dead, or B: your battery is worn out. If your Ds lite still does the same thing after you have let the battery FULLY charge before using it, then your battery is most likely the problem. If this is the case, then you need to buy a new battery, which is between $6 - $10 at some places. When you get your new battery (if you buy one), you need to first let the Ds lite charge COMPLETELY before use, and then after that let it fully (or at least mostly) die before charging it again. If you constantly charge it when it is half or mostly full, then than will MAJORLY shorten the battery life, and eventually cause the problem you described. You need to take care of the battery or it wont last very long.