Low voltage on 120V receptacle after too much load
Since you have a meter, flip the breaker off for this circuit and remove the receptacle. Keep the wires separated and safe and then flip the breaker back on. Check the voltage between the hot and neutral. If it's 120V, then replace the receptacle. If it's still 13V, then the problem is upstream of this between the walls or at another receptacle along the line. 

If this is the issue, you would have to check out each receptacle along the line until it returns to 120V from hot to neutral, this would mean the problem is between that receptacle and the last measured. Since you said that the nearest receptacle is at 120V, I'm assuming that it's on the same circuit and the next wired in sequence? If anything, your problem is between this receptacle and your problem receptacle; in wiring or in devices.