Better late than never, here is a correct answer. (I hope.)
Your MagSafe power supply's identification chip has fried, or MagSafe plug is otherwise damaged. Inside every Apple MagSafe tip/plug/outlet is a microchip which does two things: it IDs itself (as 45W/60W/85W/Airline and serial number) and controls red/green LEDs. Your MagSafe plug is damaged: maybe its chip is fried/zapped, maybe cracked from severe traumatic impact (not very likely), maybe damaged by liquid intrusion. I can believe in a short circuit: maybe minerals dried in a conductive path to LED, so that is why it is always illuminated.
What to do? Keep your "power brick" AC adapter, get a replacement MagSafe plug or cable assembly, and splice it on.
If you cannot find a used Original Apple MagSafe tip, then a non-authorised replica is acceptable. One will be better-off with a counterfeit tip spliced onto genuine power supply, than a knock-off power supply.
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very similar problem, cross reference: [post|467199]
Better late than never, here is a correct answer. (I hope.)
Your MagSafe power supply's identification chip has fried, or MagSafe plug is otherwise damaged. Inside every Apple MagSafe tip/plug/outlet is a microchip which does two things: it IDs itself (as 45W/60W/85W/Airline and serial number) and controls red/green LEDs. Your MagSafe plug is damaged: maybe its chip is fried/zapped, maybe cracked from severe traumatic impact (not very likely), maybe damaged by liquid intrusion. I can believe in a short circuit: maybe minerals dried in a conductive path to LED, so that is why it is always illuminated.
What to do? Keep your "power brick" AC adapter, get a replacement MagSafe plug or cable assembly, and splice it on.
If you cannot find a used Original Apple MagSafe tip, then a non-authorised replica is acceptable. One will be better-off with a counterfeit tip spliced onto genuine power supply, than a knock-off power supply.