It's not very likely that the starter pack transistor will be what you need but as yet your question is not clear.
What exactly are the "led lights" you have that need 24V? How much current do they require? Do you just need to switch them on/off or is there more control required, like brightness or colours?
BTW just being picky but an Arduino can't power 24V lights. It can control them with suitable add-ons but you definitely need a separate supply to power them.
Steve