I'd say there are two possibilities.
First, please check the original unit. In addition to the batteries and the LEDs, there ought to be a resistor. This will limit the LED current to safe levels. Without any resistor at all, it's entirely possible that you've simply destroyed one or more LEDs by hooking them directly to the battery.
The second possibility has to do with your soldering technique. What size soldering iron do you have and how long did it take you to remove and resolder the LEDs? It is entirely possible that, if you used a high-wattage soldering iron, and heated the LED leads for too long that you have just cooked one or more LED.
ETA - About choosing resistors. You did not identify the batteries you use, so there is no way to know what voltage they provide. In fact, there is no guarantee that the five LEDs are not connected in parallel.
But here's the general procedure for choosing resistors. Find the battery voltage. Then figure out the operating voltage and current requirements for the LEDs. In the case of red LEDs, 1.5 volts is a good start, and 10 to 20 mA for the current. When the LED is operating, the battery voltage will be split between the resistor and the LED. Let's say that the battery puts out 2 volts, and the LED needs 1.5. Then the difference (2 - 1.5) will appear across the resistor. The relationship between voltage and current in a resistor is called Ohm's Law, and is V = iR, where V is voltage, I is current (in amps) and R is resistance in ohms. In this case, assuming 20 mA for the LED current, the current through the resistor will also be 20 mA, since the two are in series. Then 0.5 = .020 x R, and R = .5 / .020, or 25 ohms.