You need to know the current required for the LED's which are current driven. One diode usually has about 0.7 volt drop so if they are rated at 2.2 volts there is probably a series resistor in there to limit current under some limit to keep from destroying the LED.
You can see how well they work if you can use some kind of variable voltage supply.
The LED's usually run on about 30 ma or so per bulb, so if they are in series, you have to multiply the voltage by the number in series. The ones in parallel would have a higher current total but each lamp would still use the same current, about 30 ma. So in the triple circuit, there would be about 90 ma flowing. You might want to just put those three in series so you will only have one path with 30 ma or so flowing, which would simplify the voltage and resistor situation. If you can put them in series that is, could be the wiring is wrapped up inside cloth or somehow hidden but if you can get to the wires, you can certainly put them in series then just add up all the 2.2 volts of the total, 10? which would need 22 volts to run. Or you could put the whole lot in parallel and go with just 3 volts or so at 30 odd ma times the number of bulbs, say 10 bulbs in parallel, would be about 1/3 amp, ~300 ma, total about 1 watt, not much power required.
If they are in parallel, a single bulb opening up will not stop all the lights. That is how christmas tree lights work, all in parallel so if one bulb burns out, the rest still light up.
The downside would be if the bulb shorts out, that would drop everything but usually they open up, with a high resistance.