There's nothing wrong with the circuit..it functions as designed by Scott...I've built several for charging lipo batteries and its a characteristic that the battery never appears full because the charging regime reverts to constant voltage with an end point of 4.22v/cell....so a small current will flow...but not enough for the led to remain lit...
You could add some sort of detection circuitry around the current sensing resistor to turn off charging but is it worth it...I think not..just disconnect when the led goes out