No red light when you plug it in is bad. Combined with a sudden shutdown it's very bad.
Just to check: the charger works with another phone, you've tried a different cable, and you've checked for dirt in the charging port? None of these would explain the sudden shutdown, but just want to be sure there isn't a simple/inexpensive solution to the lack of charging.
Unfortunately I think the most likely explanation for what you've seen (sudden shutdown and lack of charging light) is a motherboard failure. Maybe a catastrophic battery failure, but motherboard would be my guess. Either would require repair or replacement, so the only real difference is the cost of repair.
Sorry, but I don't think there's anything you can do yourself.