Diagnosing potential damage to garden solar light
These boards have five parts. The led, a axial inductor (the green resistor looking thing, likely 100 milli Henry), the solar panel, a NiMH battery, and the IC (Chip on Blob, it's just a common solar panel led boost driver).

Too much heat could certainly have damaged any of these parts. Assuming you disconnected the battery before soldering, the most likely part damaged is the IC bond wires. The inductor is easy to test, just measure it's resistance. This would be enough to know if it is open or not.  Use the good boards inductor as a reference. 

Other things to check. Battery should be above 0.8 Volts or the IC won't turn the led on. It's a under voltage lockout to prevent the battery from being drain to death. Measure the solar panel under light and load, like 470 ohm resistor. It should provide 2+ volts under full light. Compare that to the working boards battery and panel.

Consider it dead, it only cost you a dollar or so.