Mass produced devices like that are usually cheaper to replace then to spend time and money trying to repair them. Are the batteries even still good? Chances are that buying new batteries will cost more than the complete device.
Manufacturers of cheaply made devices like that seldom provide any schematics so the only way of knowing the value of the resistor would be to find an identical device and open it up and see. Or you could draw a schematic based on the board that you have there, then post it here and someone could give you an educated guess as the value of the burnt resistor. As I said in the first paragraph, it's probably more trouble than it's worth.