LED and light bulb in series

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.

It has been posted elsewhere, but there appear to be several reasons why LEDs can glow when the light is switched off. The underlying reason, of course, is that LEDs require very little power to illuminate. There is a video of someone standing under some transmissions lines and getting a LED to glow from the induced current.

1. The switch is not completely switching off; it is a dimmer or some other device that leaks some current when it is supposed to be off.

2. The switch is installed on the neutral wire. In this case, it isn't even necessary to have any leakage to earth. The cabling and circuitry acts as a capacitor that can have sufficient capacitance to allow a small current to flow.

3. There isn't a neutral wire. Though most countries have a neutral wire at near ground, there are some exceptions. Some old installations in Belgium and France have two hot wires. Hence (2) applies always.

4. If you have a dual-pole setup there are two live wires, one off at one end and one off at the other end. If the wire is long enough a current can be induced in the overlapping section of wire.