Dell Inspiron n5110 battery/charger lights goes out when plugged into laptop

That is normal behavior for the adapter if there is a short in the system - I'll suggest first putting the adapter on a multi-meter, see if it's putting out 19v (or whatever) and acting proper, in the same vain, you might check your battery's health also, does it do that when the battery is not in? If so, you'll want to pull your motherboard and go over it really good to see if there is anything obviously wrong (swollen caps, obvious short, etc...)

Just to confirm I understand this correctly, it is the adapter's light that goes out... not the charging light on the laptop (which shouldn't light at all by my reckoning)