You need to know what that switch does exactly.
Does it switch a voltage to GND ?
Or does it switch a power line (less likely) ?
Once you know that, you can figure out whether you need an extra transistor or not.
You should also consider to use high efficiency LEDs, which only need something like 2 mA instead of 20 mA, and so save you some battery time.
Thank you so much for the guidance. I really appreciate it. This will get me headed in the right direction. I really like the idea of having my player glow blue through the window in the lid while it is playing. Knowing that it is possible gives me hope I will eventually get it figured out.
Oh, and Boffin, thanks for absolutely nothing! So glad you could grace us with your presence regardless. We're not worthy!